ON LANDAU DAMPING - Cédric Villani LANDAU DAMPING C. MOUHOT AND C. VILLANI Abstract. Going beyond the linearized study has been a longstanding problem in the theory of Landau damping.Published

ON LANDAU DAMPING

C. MOUHOT AND C. VILLANI

Abstract. Going beyond the linearized study has been a longstanding problemin the theory of Landau damping. In this paper we establish exponential Lan-dau damping in analytic regularity. The damping phenomenon is reinterpreted interms of transfer of regularity between kinetic and spatial variables, rather thanexchanges of energy; phase mixing is the driving mechanism. The analysis involvesnew families of analytic norms, measuring regularity by comparison with solutionsof the free transport equation; new functional inequalities; a control of nonlinearechoes; sharp “deflection” estimates; and a Newton approximation scheme. Ourresults hold for any potential no more singular than Coulomb or Newton interac-tion; the limit cases are included with specific technical effort. As a side result, thestability of homogeneous equilibria of the nonlinear Vlasov equation is establishedunder sharp assumptions. We point out the strong analogy with the KAM theory,and discuss physical implications.

Contents

1. Introduction to Landau damping 42. Main result 133. Linear damping 264. Analytic norms 375. Deflection estimates 656. Bilinear regularity and decay estimates 727. Control of the time-response 838. Approximation schemes 1159. Local in time iteration 12210. Global in time iteration 12611. Coulomb/Newton interaction 15912. Convergence in large time 16513. Non-analytic perturbations 16814. Expansions and counterexamples 17215. Beyond Landau damping 179Appendix 181References 185

Keywords. Landau damping; plasma physics; galactic dynamics; Vlasov-Poissonequation.

AMS Subject Classification. 82C99 (85A05, 82D10)

1

2 C. MOUHOT AND C. VILLANI

Landau damping may be the single most famous mystery of classical plasmaphysics. For the past sixty years it has been treated in the linear setting at variousdegrees of rigor; but its nonlinear version has remained elusive, since the only avail-able results [13, 40] prove the existence of some damped solutions, without tellinganything about their genericity.

In the present work we close this gap by treating the nonlinear version of Landaudamping in arbitrarily large times, under assumptions which cover both attractiveand repulsive interactions, of any regularity down to Coulomb/Newton.

This will lead us to discover a distinctive mathematical theory of Landau damping,complete with its own functional spaces and functional inequalities. Let us makeit clear that this study is not just for the sake of mathematical rigor: indeed, weshall get new insights in the physics of the problem, and identify new mathematicalphenomena.

The plan of the paper is as follows.In Section 1 we provide an introduction to Landau damping, including historical

comments and a review of the existing literature. Then in Section 2, we state andcomment on our main result about “nonlinear Landau damping” (Theorem 2.6).

In Section 3 we provide a rather complete treatment of linear Landau damping,slightly improving on the existing results both in generality and simplicity. Thissection can be read independently of the rest.

In Section 4 we define the spaces of analytic functions which are used in theremainder of the paper. The careful choice of norms is one of the keys of ouranalysis; the complexity of the problem will naturally lead us to work with normshaving up to 5 parameters. As a first application, we shall revisit linear Landaudamping within this framework.

In Sections 5 to 7 we establish four types of new estimates (deflection estimates,short-term and long-term regularity extortion, echo control); these are the key sec-tions containing in particular the physically relevant new material.

In Section 8 we adapt the Newton algorithm to the setting of the nonlinear Vlasovequation. Then in Sections 9 to 11 we establish some iterative estimates along thisscheme. (Section 11 is devoted specifically to a technical refinement allowing tohandle Coulomb/Newton interaction.)

From these estimates our main theorem is easily deduced in Section 12.An extension to non-analytic perturbations is presented in Section 13.Some counterexamples and asymptotic expansions are studied in Section 14.Final comments about the scope and range of applicability of these results are

provided in Section 15.

ON LANDAU DAMPING 3

Even though it basically proves one main result, this paper is very long. This isdue partly to the intrinsic complexity and richness of the problem, partly to the needto develop an adequate functional theory from scratch, and partly to the inclusionof remarks, explanations and comments intended to help the reader understand theproof and the scope of the results. The whole process culminates in the extremelytechnical iteration performed in Sections 10 and 11. A short summary of our resultsand methods of proofs can be found in the expository paper [67].

This project started from an unlikely conjunction of discussions of the authorswith various people, most notably Yan Guo, Dong Li, Freddy Bouchet and EtienneGhys. We also got crucial inspiration from the books [9, 10] by James Binney andScott Tremaine; and [2] by Serge Alinhac and Patrick Gerard. Warm thanks toJulien Barre, Jean Dolbeault, Thierry Gallay, Stephen Gustafson, Gregory Ham-mett, Donald Lynden-Bell, Michael Sigal, Eric Sere and especially Michael Kiesslingfor useful exchanges and references; and to Francis Filbet and Irene Gamba forproviding numerical simulations. We are also grateful to Patrick Bernard, FreddyBouchet, Emanuele Caglioti, Yves Elskens, Yan Guo, Zhiwu Lin, Michael Loss, PeterMarkowich, Govind Menon, Yann Ollivier, Mario Pulvirenti, Jeff Rauch, Igor Rod-nianski, Peter Smereka, Yoshio Sone, Tom Spencer, and the team of the PrincetonPlasma Physics Laboratory for further constructive discussions about our results.Finally, we acknowledge the generous hospitality of several institutions: Brown Uni-versity, where the first author was introduced to Landau damping by Yan Guo inearly 2005; the Institute for Advanced Study in Princeton, who offered the secondauthor a serene atmosphere of work and concentration during the best part of thepreparation of this work; Cambridge University, who provided repeated hospitalityto the first author thanks to the Award No. KUK-I1-007-43, funded by the KingAbdullah University of Science and Technology (KAUST); and the University ofMichigan, where conversations with Jeff Rauch and others triggered a significantimprovement of our results.

Our deep thanks go to the referees for their careful examination of the manu-script. We dedicate this paper to two great scientists who passed away during theelaboration of our work. The first one is Carlo Cercignani, one of the leaders of ki-netic theory, author of several masterful treatises on the Boltzmann equation, and along-time personal friend of the second author. The other one is Vladimir Arnold, amathematician of extraordinary insight and influence; in this paper we shall uncovera tight link between Landau damping and the theory of perturbation of completelyintegrable Hamiltonian systems, to which Arnold has made major contributions.


1. Introduction to Landau damping

1.1. Discovery. Under adequate assumptions (collisionless regime, nonrelativisticmotion, heavy ions, no magnetic field), a dilute plasma is well described by thenonlinear Vlasov–Poisson equation

(1.1)∂f

∂t+ v · ∇xf +

F

m· ∇vf = 0,

where f = f(t, x, v) ≥ 0 is the density of electrons in phase space (x= position,v= velocity), m is the mass of an electron, and F = F (t, x) is the mean-field (self-consistent) electrostatic force:

(1.2) F = −eE, E = ∇∆−1(4πρ).

Here e > 0 is the absolute electron charge, E = E(t, x) is the electric field, andρ = ρ(t, x) is the density of charges

(1.3) ρ = ρi − e

∫f dv,

ρi being the density of charges due to ions. This model and its many variants are oftantamount importance in plasma physics [1, 5, 47, 52].

In contrast to models incorporating collisions [92], the Vlasov–Poisson equation istime-reversible. However, in 1946 Landau [50] stunned the physical community bypredicting an irreversible behavior on the basis of this equation. This “astonishingresult” (as it was called in [86]) relied on the solution of the Cauchy problem forthe linearized Vlasov–Poisson equation around a spatially homogeneous Maxwellian(Gaussian) equilibrium. Landau formally solved the equation by means of Fourierand Laplace transforms, and after a study of singularities in the complex plane,concluded that the electric field decays exponentially fast; he further studied the rateof decay as a function of the wave vector k. Landau’s computations are reproducedin [52, Section 34] or [1, Section 4.2].

An alternative argument appears in [52, Section 30]: there the thermodynamicalformalism is used to compute the amount of heat Q which is dissipated when a(small) oscillating electric field E(t, x) = E ei(k·x−ωt) (k a wave vector, ω > 0 afrequency) is applied to a plasma whose distribution f 0 is homogeneous in spaceand isotropic in velocity space; the result is

(1.4) Q = −|E|2 πme2ω

|k|2 φ′

(ω

|k|

),

ON LANDAU DAMPING 5

where φ(v1) =∫f 0(v1, v2, v3) dv2 dv3. In particular, (1.4) is always positive (see the

last remark in [52, Section 30]), which means that the system reacts against theperturbation, and thus possesses some “active” stabilization mechanism.

A third argument [52, Section 32] consists in studying the dispersion relation,or equivalently searching for the (generalized) eigenmodes of the linearized Vlasov–Poisson equation, now with complex frequency ω. After appropriate selection, theseeigenmodes are all decaying (ℑm ω < 0) as t → ∞. This again suggests stability,although in a somewhat weaker sense than the computation of heat release.

The first and third arguments also apply to the gravitational Vlasov–Poisson equa-tion, which is the main model for nonrelativistic galactic dynamics. This equationis similar to (1.1), but now m is the mass of a typical star (!), and f is the densityof stars in phase space; moreover the first equation of (1.2) and the relation (1.3)should be replaced by

(1.5) F = −GmE, ρ = m

∫f dv;

where G is the gravitational constant, E the gravitational field, and ρ the densityof mass. The books by Binney and Tremaine [9, 10] constitute excellent referencesabout the use of the Vlasov–Poisson equation in stellar dynamics — where it isoften called the “collisionless Boltzmann equation”, see footnote on [10, p. 276]. On“intermediate” time scales, the Vlasov–Poisson equation is thought to be an accuratedescription of very large star systems [28], which are now accessible to numericalsimulations.

Since the work of Lynden-Bell [56] it has been recognized that Landau damping,and wilder collisionless relaxation processes generically dubbed “violent relaxation”,constitute a fundamental stabilizing ingredient of galactic dynamics. Without thesestill poorly understood mechanisms, the surprisingly short time scales for relaxationof the galaxies would remain unexplained.

One main difference between the electrostatic and the gravitational interactionsis that in the latter case Landau damping should occur only at wavelengths smallerthan the Jeans length [10, Section 5.2]; beyond this scale, even for Maxwellianvelocity profiles, the Jeans instability takes over and governs planet and galaxyaggregation.1

1or at least would do, if galactic matter was smoothly distributed; in presence of “microscopic”heterogeneities, a phase transition for aggregation can occur far below this scale [45]. In thelanguage of statistical mechanics, the Jeans length corresponds to a “spinodal point” rather thana phase transition [85].


On the contrary, in (classical) plasma physics, Landau damping should hold atall scales under suitable assumptions on the velocity profile; and in fact one is ingeneral not interested in scales smaller than the Debye length, which is roughlydefined in the same way as the Jeans length.

Nowadays, not only has Landau damping become a cornerstone of plasma physics2,but it has also made its way in other areas of physics (astrophysics, but also windwaves, fluids, superfluids,. . . ) and even biophysics. One may consult the concisesurvey papers [74, 79, 91] for a discussion of its influence and some applications.

1.2. Interpretation. True to his legend, Landau deduced the damping effect froma mathematical-style study3, without bothering to give a physical explanation of theunderlying mechanism. His arguments anyway yield exact formulas, which in prin-ciple can be checked experimentally, and indeed provide good qualitative agreementwith observations [57].

A first set of problems in the interpretation is related to the arrow of time. Inthe thermodynamic argument, the exterior field is awkwardly imposed from time−∞ on; moreover, reconciling a positive energy dissipation with the reversibilityof the equation is not obvious. In the dispersion argument, one has to arbitrarilyimpose the location of the singularities taking into account the arrow of time (viathe Plemelj formula); then the spectral study requires some thinking. All in all, themost convincing argument remains Landau’s original one, since it is based only onthe study of the Cauchy problem, which makes more physical sense than the studyof the dispersion relation (see the remark in [9, p. 682]).

A more fundamental issue resides in the use of analytic function theory, withcontour integration, singularities and residue computation, which has played a majorrole in the theory of the Vlasov–Poisson equation ever since Landau [52, Chapter32] [10, Subsection 5.2.4] and helps little, if at all, to understand the underlyingphysical mechanism.4

The most popular interpretation of Landau damping considers the phenomenonfrom an energetic point of view, as the result of the interaction of a plasma wave

2Ryutov [79] estimated in 1998 that “approximately every third paper on plasma physics andits applications contains a direct reference to Landau damping”.

3not completely rigorous from the mathematical point of view, but formally correct, in contrastto the previous studies by Landau’s fellow physicists — as Landau himself pointed out withoutmercy [50].

4Van Kampen [90] summarizes the conceptual problems posed to his contemporaries by Landau’streatment, and comments on more or less clumsy attempts to resolve the apparent paradox causedby the singularity in the complex plane.

ON LANDAU DAMPING 7

with particles of nearby velocity [87, p. 18] [10, p. 412] [1, Section 4.2.3] [52, p. 127].In a nutshell, the argument says that dominant exchanges occur with those particleswhich are “trapped” by the wave because their velocity is close to the wave velocity.If the distribution function is a decreasing function of |v|, among trapped particlesmore are accelerated than are decelerated, so the wave loses energy to the plasma— or the plasma surfs on the wave — and the wave is damped by the interaction.

Appealing as this image may seem, to a mathematically-oriented mind it willprobably make little sense at first hearing.5 A more down-to-Earth interpretationemerged in the fifties from the “wave packet” analysis of Van Kampen [90] and Case[14]: Landau damping would result from phase mixing. This phenomenon, well-known in galactic dynamics, describes the damping of oscillations occurring whena continuum is transported in phase space along an anharmonic Hamiltonian flow[10, pp. 379–380]. The mixing results from the simple fact that particles followingdifferent orbits travel at different angular6 speeds, so perturbations start “spiralling”(see Figure 4.27 on [10, p. 379]) and homogenize by fast spatial oscillation. Fromthe mathematical point of view, phase mixing results in weak convergence; from thephysical point of view, this is just the convergence of observables, defined as averagesover the velocity space (this is sometimes called “convergence in the mean”).

At first sight, both points of view seem hardly compatible: Landau’s scenariosuggests a very smooth process, while phase mixing involves tremendous oscillations.The coexistence of these two interpretations did generate some speculation on thenature of the damping, and on its relation to phase mixing, see e.g. [44] or [10,p. 413]. There is actually no contradiction between the two points of view: manyphysicists have rightly pointed out that that Landau damping should come withfilamentation and oscillations of the distribution function [90, p. 962] [52, p. 141] [1,Vol. 1, pp. 223–224] [55, pp. 294–295]. Nowadays these oscillations can be visualizedspectacularly thanks to deterministic numerical schemes, see e.g. [95] [38, Fig. 3][27]. We reproduce below some examples provided by Filbet.

In any case, there is still no definite interpretation of Landau damping: as noted byRyutov [79, Section 9], papers devoted to the interpretation and teaching of Landaudamping were still appearing regularly fifty years after its discovery; to quote justa couple of more recent examples let us mention works by Elskens and Escande[23, 24, 25]. The present paper will also contribute a new point of view.

5Escande [25, Chapter 4, Footnote 6] points out some misconceptions associated with the surferimage.

6“Angular” here refers to action-angle variables, and applies even for straight trajectories in atorus.


-4e-05

-3e-05

-2e-05

-1e-05

0

1e-05

2e-05

3e-05

4e-05

-6 -4 -2 0 2 4 6

h(v)

v

t = 0.16

-4e-05

-3e-05

-2e-05

-1e-05

0

1e-05

2e-05

3e-05

4e-05

-6 -4 -2 0 2 4 6

h(v)

v

t = 2.00

Figure 1. A slice of the distribution function (relative to a homoge-neous equilibrium) for gravitational Landau damping, at two differenttimes.

-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

0 10 20 30 40 50 60 70 80 90 100

log(

Et(

t))

t

electric energy in log scale

-23

-22

-21

-20

-19

-18

-17

-16

-15

-14

-13

-12

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

log(

Et(

t))

t

gravitational energy in log scale

Figure 2. Time-evolution of the norm of the field, for electrostatic(on the left) and gravitational (on the right) interactions. Notice thefast Langmuir oscillations in the electrostatic case.

1.3. Range of validity. The following issues are addressed in the literature [41,44, 59, 95] and slightly controversial:

• Does Landau damping really hold for gravitational interaction? The case seemsthinner in this situation than for plasma interaction, all the more that there aremany instability results in the gravitational context; up to now there has been noconsensus among mathematical physicists [77]. (Numerical evidence is not conclusive

ON LANDAU DAMPING 9

because of the difficulty of accurate simulations in very large time — even in onedimension of space.)

• Does the damping hold for unbounded systems? Counterexamples from [30, 31]show that some kind of confinement is necessary, even in the electrostatic case.More precisely, Glassey and Schaeffer show that a solution of the linearized Vlasov–Poisson equation in the whole space (linearized around a homogeneous equilibriumf 0 of infinite mass) decays at best like O(t−1), modulo logarithmic corrections, forf 0(v) = c/(1 + |v|2); and like O((log t)−α) if f 0 is a Gaussian. In fact, Landau’soriginal calculations already indicated that the damping is extremely weak at largewavenumbers; see the discussion in [52, Section 32]. Of course, in the gravitationalcase, this is even more dramatic because of the Jeans instability.

• Does convergence hold in infinite time for the solution of the “full” nonlinearequation? This is not clear at all since there is no mechanism that would keep thedistribution close to the original equilibrium for all times. Some authors do not be-lieve that there is convergence as t→ ∞; others believe that there is convergence butargue that it should be very slow [41], say O(1/t). In the first mathematically rig-orous study of the subject, Backus [4] notes that in general the linear and nonlinearevolution break apart after some (not very large) time, and questions the validityof the linearization.7 O’Neil [73] argues that relaxation holds in the “quasilinearregime” on larger time scales, when the “trapping time” (roughly proportional theinverse square root of the size of the perturbation) is much smaller than the dampingtime. Other speculations and arguments related to trapping appear in many sources,e.g. [59, 62]. Kaganovich [43] argues that nonlinear effects may quantitatively affectLandau damping related phenomena by several orders of magnitude.

The so-called “quasilinear relaxation theory” [52, Section 49] [1, Section 9.1.2][47, Chapter 10] uses second-order approximation of the Vlasov equation to predictthe convergence of the spatial average of the distribution function. The procedure ismost esoteric, involving averaging over statistical ensembles, and diffusion equationswith discontinuous coefficients, acting only near the resonance velocity for particle-wave exchanges. Because of these discontinuities, the predicted asymptotic stateis discontinuous, and collisions are invoked to restore smoothness. Linear Fokker–Planck equations8 in velocity space have also been used in astrophysics [56, p. 111],

7From the abstract: “The linear theory predicts that in stable plasmas the neglected term willgrow linearly with time at a rate proportional to the initial disturbance amplitude, destroying thevalidity of the linear theory, and vitiating positive conclusions about stability based on it.”

8These equations act on some ensemble average of the distribution; they are different from theVlasov–Landau equation.


but only on phenomenological grounds (the ad hoc addition of a friction term leadingto a Gaussian stationary state); and this procedure has been exported to the studyof two-dimensional incompressible fluids [15, 16].

Even if it were more rigorous, quasilinear theory only aims at second-order cor-rections, but the effect of higher order perturbations might be even worse. Thinkof something like e−t

∑n(εntn)/

√n!, then truncation at any order in ε converges

exponentially fast as t→ ∞, but the whole sum diverges to infinity.Careful numerical simulation [95] seems to show that the solution of the nonlinear

Vlasov–Poisson equation does converge to a spatially homogeneous distribution,but only as long as the size of the perturbation is small enough. We shall call thisphenomenon nonlinear Landau damping. This terminology summarizes well theproblem, still it is subject to criticism since (a) Landau himself sticked to the linearcase and did not discuss the large-time convergence of the distribution function; (b)damping is expected to hold when the regime is close to linear, but not necessarilywhen the nonlinear term dominates9; and (c) this expression is also used to designaterelated but different phenomena [1, Section 10.1.3]. It should be kept in mind that inthe present paper, nonlinearity does manifest itself, not because there is a significantinitial departure from equilibrium (our initial data will be very close to equilibrium),but because we are addressing very large times, and this is all the more tricky tohandle that the problem is highly oscillating.

• Is Landau damping related to the more classical notion of stability in orbitalsense? Orbital stability means that the system, slightly perturbed at initial timefrom an equilibrium distribution, will always remain close to this equilibrium. Evenin the favorable electrostatic case, stability is not granted; the most prominentphenomenon being the Penrose instability [75] according to which a distribution withtwo deep bumps may be unstable. In the more subtle gravitational case, variousstability and instability criteria are associated with the names of Chandrasekhar,Antonov, Goodman, Doremus, Feix, Baumann, . . . [10, Section 7.4]. There is awidespread agreement (see e.g. the comments in [95]) that Landau damping andstability are related, and that Landau damping cannot be hoped for if there is noorbital stability.

1.4. Conceptual problems. Summarizing, we can identify three main conceptualobstacles which make Landau damping mysterious, even sixty years after its discov-ery:

9although phase mixing might still play a crucial role in violent relaxation or other unclassifiednonlinear phenomena.

ON LANDAU DAMPING 11

(i) The equation is time-reversible, yet we are looking for an irreversible behavioras t→ +∞ (or t→ −∞). The value of the entropy does not change in time, whichphysically speaking means that there is no loss of information in the distributionfunction. The spectacular experiment of the “plasma echo” illustrates this conser-vation of microscopic information [32, 58]: a plasma which is apparently back toequilibrium after an initial disturbance, will react to a second disturbance in a waythat shows that it has not forgotten the first one.10 And at the linear level, if thereare decaying modes, there also has to be growing modes!

(ii) When one perturbs an equilibrium, there is no mechanism forcing the systemto go back to this equilibrium in large time; so there is no justification in the use oflinearization to predict the large-time behavior.

(iii) At the technical level, Landau damping (in Landau’s own treatment) restson analyticity, and its most attractive interpretation is in terms of phase mixing.But both phenomena are incompatible in the large-time limit: phase mixing impliesan irreversible deterioration of analyticity. For instance, it is easily checked thatfree transport induces an exponential growth of analytic norms as t→ ∞ — exceptif the initial datum is spatially homogeneous. In particular, the Vlasov–Poissonequation is unstable (in large time) in any norm incorporating velocity regularity.(Space-averaging is one of the ingredients used in the quasilinear theory to formallyget rid of this instability.)

How can we respond to these issues?One way to solve the first problem (time-reversibility) is to appeal to Van Kam-

pen modes as in [10, p. 415]; however these are not so physical, as noticed in [9,p. 682]. A simpler conceptual solution is to invoke the notion of weak convergence:reversibility manifests itself in the conservation of the information contained in thedensity function; but information may be lost irreversibly in the limit when we con-sider weak convergence. Weak convergence only describes the long-time behaviorof arbitrary observables, each of which does not contain as much information asthe density function.11 As a very simple illustration, consider the time-reversibleevolution defined by u(t, x) = eitxui(x), and notice that it does converge weakly to0 as t → ±∞; this convergence is even exponentially fast if the initial datum ui is

10Interestingly enough, this experiment was suggested as a way to evaluate the strength ofirreversible phenomena going on inside a plasma, e.g. the collision frequency, by measuring at-tenuations with respect to the predicted echo. See [84] for an interesting application and strikingpictures.

11In Lynden-Bell’s appealing words [55, p.295], “a system whose density has achieved a steadystate will have information about its birth still stored in the peculiar velocities of its stars.”


analytic. (Our example is not chosen at random: although it is extremely simple,it may be a good illustration of what happens in phase mixing.) In a way, microso-copic reversibility is compatible with macroscopic irreversibility, provided that the“microscopic regularity” is destroyed asymptotically.

Still in respect to this reversibility, it should be noted that the “dual” mechanismof radiation, according to which an infinite-dimensional system may lose energytowards very large scales, is relatively well understood and recognized as a crucialstability mechanism [3, 83].

The second problem (lack of justification of the linearization) only indicates thatthere is a wide gap between the understanding of linear Landau damping, and thatof the nonlinear phenomenon. Even if unbounded corrections appear in the lin-earization procedure, the effect of the large terms might be averaged over time orother variables.

The third problem, maybe the most troubling from an analyst’s perspective, doesnot dismiss the phase mixing explanation, but suggests that we shall have to keeptrack of the initial time, in the sense that a rigorous proof cannot be based onthe propagation of some phenomenon. This situation is of course in sharp contrastwith the study of dissipative systems possessing a Lyapunov functional, as do manycollisional kinetic equations [92, 93]; it will require completely different mathematicaltechniques.

1.5. Previous mathematical results. At the linear level, the first rigorous treat-ments of Landau damping were performed in the sixties; see Saenz [80] for rathercomplete results and a review of earlier works. The theory was rediscovered andrenewed at the beginning of the eighties by Degond [20], and Maslov and Fedoryuk[61]. In all these works, analytic arguments play a crucial role (for instance for theanalytic extension of resolvent operators), and asymptotic expansions for the electricfield associated to the linearized Vlasov–Poisson equation are obtained.

Also at the linearized level, there are counterexamples by Glassey and Schaeffer[30, 31] showing that there is in general no exponential decay for the linearizedVlasov–Poisson equation without analyticity, or without confining.

In a nonlinear setting, the only rigorous treatments so far are those by Caglioti–Maffei [13], and later Hwang–Velazquez [40]. Both sets of authors work in the one-dimensional torus and use fixed-point theorems and perturbative arguments to provethe existence of a class of analytic solutions behaving, asymptotically as t → +∞,and in a strong sense, like perturbed solutions of free transport. Since solutions offree transport weakly converge to spatially homogeneous distributions, the solutionsconstructed by this “scattering” approach are indeed damped. The weakness of


these results is that they say nothing about the initial perturbations leading to suchsolutions, which could be very special. In other words: damped solutions do exist,but do we ever reach them?

Sparse as it may seem, this list is kind of exhaustive. On the other hand, thereis a rather large mathematical literature on the orbital stability problem, due toGuo, Rein, Strauss, Wolansky and Lemou–Mehats–Raphael. In this respect seefor instance [35] for the plasma case, and [34, 51] for the gravitational case; thesesources contain many references on the subject. This body of works has confirmedthe intuition of physicists, although with quite different methods. The gap betweena formal, linear treatment and a rigorous, nonlinear one is striking: Compare theAppendix of [34] to the rest of the paper. In the gravitational case, these works donot consider homogeneous equilibria, but only localized solutions.

Our treatment of Landau damping will be performed from scratch, and will notrely on any of these results.

2. Main result

2.1. Modelling. We shall work in adimensional units throughout the paper, in ddimensions of space and d dimensions of velocity (d ∈ N).

As should be clear from our presentation in Section 1, to observe Landau damping,we need to put a restriction on the length scale (anyway plasmas in experiments areusually confined). To achieve this we shall take the position space to be the d-dimensional torus of sidelength L, namely Td

L = Rd/(LZ)d. This is admittedly a bitunrealistic, but it is commonly done in plasma physics (see e.g. [5]).

In a periodic setting the Poisson equation has to be reinterpreted, since ∆−1ρ isnot well-defined unless

∫Td

Lρ = 0. The natural solution consists in removing the

mean value of ρ, independently of any “neutrality” assumption. Let us sketch ajustification in the important case of Coulomb interaction: due to the screeningphenomenon, we may replace the Coulomb potential V by a potential Vκ exhibitinga “cutoff” at large distances (typically Vκ could be of Debye type [5]; anyway thechoice of approximation has no influence on the result.) If ∇Vκ ∈ L1(Rd), then∇Vκ ∗ ρ makes sense for a periodic ρ, and moreover

(∇Vκ ∗ ρ)(x) =

∫

Rd

∇Vκ(x− y) ρ(y) dy =

∫

[0,L]d∇V (L)

κ (x− y) ρ(y) dy,


where V(L)κ (z) =

∑ℓ∈Zd Vκ(z + ℓL). Passing to the limit as κ→ 0 yields

∫

[0,L]d∇V (L)(x−y) ρ(y) dy =

∫

[0,L]d∇V (L)(x−y)

(ρ−〈ρ〉

)(y) dy = −∇∆−1

L

(ρ−〈ρ〉

),

where ∆−1L is the inverse Laplace operator on Td

L.In the case of galactic dynamics there is no screening; however it is customary to

remove the zeroth order term of the density. This is known as the Jeans swindle,a trick considered as efficient but logically absurd. In 2003, Kiessling [46] re-openedthe case and acquitted Jeans, on the basis that his “swindle” can be justified by asimple limit procedure, similar to the one presented above; however, the physicalbasis for the limit is less transparent and subject to debate. For our purposes, itdoes not matter much: since anyway periodic boundary conditions are not realisticin a cosmological setting, we may just as well say that we adopt the Jeans swindleas a simple phenomenological model.

More generally, we may consider any interaction potential W on TdL, satisfying

the natural symmetry assumption W (−z) = W (z) (that is, W is even), as well ascertain regularity assumptions. Then the self-consistent field will be given by

F = −∇W ∗ ρ, ρ(x) =

∫f(x, v) dv,

where now ∗ denotes the convolution on TdL.

In accordance with our conventions from Appendix A.3, we shall write W (L)(k) =∫Td

Le−2iπk· x

L W (x) dx. In particular, if W is the periodization of a potential Rd → R

(still denoted W by abuse of notation), i.e.,

W (x) = W (L)(x) =∑

ℓ∈Zd

W (x+ ℓL),

then

(2.1) W (L)(k) = W

(k

L

),

where W (ξ) =∫

Rd e−2iπξ·xW (x) dx is the original Fourier transform in the whole

space.

2.2. Linear damping. It is well-known that Landau damping requires some stabil-ity assumptions on the unperturbed homogeneous distribution function, say f 0(v).


In this paper we shall use a very general assumption, expressed in terms of theFourier transform

(2.2) f 0(η) =

∫

Rd

e−2iπη·v f 0(v) dv,

the length L, and the interaction potential W . To state it, we define, for t ≥ 0 andk ∈ Zd,

(2.3) K0(t, k) = −4π2 W (L)(k) f 0

(kt

L

) |k|2L2

t;

and, for any ξ ∈ C, we define a function L via the following Fourier–Laplace trans-form of K0 in the time variable:

(2.4) L(ξ, k) =

∫ +∞

0

e2πξ∗|k|L

tK0(t, k) dt,

where ξ∗ is the complex conjugate to ξ. Our linear damping condition is expressedas follows:

(L) There are constants C0, λ, κ > 0 such that for any η ∈ Rd, |f 0(η)| ≤C0 e

−2πλ|η|; and for any ξ ∈ C with 0 ≤ ℜe ξ < λ,

infk∈Zd

∣∣L(ξ, k) − 1∣∣ ≥ κ.

We shall prove in Section 3 that (L) implies Landau damping. For the moment,let us give a few sufficient conditions for (L) to be satisfied. The first one canbe thought of as a smallness assumption on either the length, or the potential, orthe velocity distribution. The other conditions involve the marginals of f 0 alongarbitrary wave vectors k:

(2.5) ϕk(v) =

∫

k|k|

v+k⊥

f 0(w) dw, v ∈ R.

All studies known to us are based on one of these assumptions, so (L) appears as aunifying condition for linear Landau damping around a homogeneous equilibrium.

Proposition 2.1. Let f 0 = f 0(v) be a velocity distribution such that f 0 decaysexponentially fast at infinity, let L > 0 and let W be an even interaction potentialon Td

L, W ∈ L1(Td). If any one of the following conditions is satisfied:


(a) smallness:

(2.6) 4π2

(maxk∈Zd

∗

∣∣W (L)(k)∣∣) (

sup|σ|=1

∫ ∞

0

∣∣f 0(rσ)∣∣ r dr

)< 1;

(b) repulsive interaction and decreasing marginals: for all k ∈ Zd and v ∈ R,

(2.7) W (L)(k) ≥ 0;

v < 0 =⇒ ϕ′

k(v) > 0

v > 0 =⇒ ϕ′k(v) < 0;

(c) generalized Penrose condition on marginals: for all k ∈ Zd,

(2.8) ∀w ∈ R, ϕ′k(w) = 0 =⇒ W (L)(k)

(p.v.

∫

R

ϕ′k(v)

v − wdv

)< 1;

then (L) holds true for some C0, λ, κ > 0.

Remark 2.2. [52, Problem, Section 30] If f 0 is radially symmetric and positive,and d ≥ 3, then all marginals of f 0 are decreasing functions of |v|. Indeed, if

ϕ(v) =∫

Rd−1 f(√v2 + |w|2) dw, then after differentiation and integration by parts

we findϕ′(v) = −(d − 3) v

∫

Rd−1

f(√

v2 + |w|2) dw|w|2 (d ≥ 4)

ϕ′(v) = −2π v f(|v|) (d = 3).

Example 2.3. Take a gravitational interaction and Mawellian background:

W (k) = − Gπ |k|2 , f 0(v) = ρ0 e−

|v|2

2T

(2πT )d/2.

Recalling (2.1), we see that (2.6) becomes

(2.9) L <

√π T

G ρ0=: LJ(T, ρ0).

The length LJ is the celebrated Jeans length [10, 46], so criterion (a) can be applied,all the way up to the onset of the Jeans instability.


Example 2.4. If we replace the gravitational interaction by the electrostatic interac-tion, the same computation yields

(2.10) L <

√π T

e2 ρ0=: LD(T, ρ0),

and now LD is essentially the Debye length. Then criterion (a) becomes quiterestrictive, but because the interaction is repulsive we can use criterion (b) as soonas f 0 is a strictly monotone function of |v|; this covers in particular Maxwelliandistributions, independently of the size of the box. Criterion (b) also applies if d ≥ 3and f 0 has radial symmetry. For given L > 0, the condition (L) being open, it willalso be satisfied if f 0 is a small (analytic) perturbation of a profile satisfying (b);this includes the so-called “small bump on tail” stability. Then if the distributionpresents two large bumps, the Penrose instability will take over.

Example 2.5. For the electrostatic interaction in dimension 1, (2.8) becomes

(2.11) (f 0)′(w) = 0 =⇒∫

(f 0)′(v)

v − wdv <

π

e2 L2.

This is a variant of the Penrose stability condition [75]. This criterion is in generalsharp for linear stability (up to the replacement of the strict inequality by the non-strict one, and assuming that the critical points of f 0 are nondegenerate); see [53,Appendix] for precise statements.

We shall show in Section 3 that (L) implies linear Landau damping (Theorem3.1); then we shall prove Proposition 2.1 at the end of that section. The generalideas are close to those appearing in previous works, including Landau himself; theonly novelties lie in the more general assumptions, the elementary nature of thearguments, and the slightly more precise quantitative results.

2.3. Nonlinear damping. As others have done before in the study of Vlasov–Poisson [13], we shall quantify the analyticity by means of natural norms involvingFourier transform in both variables (also denoted with a tilde in the sequel). So wedefine

(2.12) ‖f‖λ,µ = supk,η

(e2πλ|η| e2πµ

|k|L

∣∣f (L)(k, η)∣∣),

where k varies in Zd, η ∈ Rd, λ, µ are positive parameters, and we recall the depen-dence of the Fourier transform on L (see Appendix A.3 for conventions). Now wecan state our main result as follows:


Theorem 2.6 (Nonlinear Landau damping). Let f 0 : Rd → R+ be an analyticvelocity profile. Let L > 0 and W : Td

L → R be an even interaction potentialsatisfying

(2.13) ∀ k ∈ Zd, |W (L)(k)| ≤ CW

|k|1+γ

for some constants CW > 0, γ ≥ 1. Assume that f 0 and W satisfy the stabilitycondition (L) from Subsection 2.2, with some constants λ, κ > 0; further assumethat, for the same parameter λ,

(2.14) supη∈Rd

(|f 0(η)| e2πλ|η|

)≤ C0,

∑

n∈Nd0

λn

n!‖∇n

vf0‖L1(Rd) ≤ C0 < +∞.

Then for any 0 < λ′ < λ, β > 0, 0 < µ′ < µ, there is ε = ε(d, L, CW , C0, κ, λ, λ′, µ, µ′, β, γ)

with the following property: if fi = fi(x, v) is an initial datum satisfying

(2.15) δ := ‖fi − f 0‖λ,µ +

∫∫

TdL×Rd

|fi − f 0| eβ|v| dv dx ≤ ε,

then

• the unique classical solution f to the nonlinear Vlasov equation

(2.16)∂f

∂t+ v · ∇xf − (∇W ∗ ρ) · ∇vf = 0, ρ =

∫

Rd

f dv,

with initial datum f(0, · ) = fi, converges in the weak topology as t → ±∞, withrate O(e−2πλ′|t|), to a spatially homogeneous equilibrium f±∞ (that is, it convergesto f+∞ as t→ +∞, and to f−∞ as t→ −∞).

• the distribution function composed with the backward free transport, f(t, x+vt, v),converges strongly to f±∞ as t→ ±∞;

• the density ρ(t, x) =∫f(t, x, v) dv converges in the strong topology as t → ±∞,

with rate O(e−2πλ′|t|), to the constant density

ρ∞ =1

Ld

∫

Rd

∫

TdL

fi(x, v) dx dv;

in particular the force F = −∇W ∗ ρ converges exponentially fast to 0;

• the space average 〈f〉(t, v) =∫f(t, x, v) dx converges in the strong topology as

t→ ±∞, with rate O(e−2πλ′|t|), to f±∞.


More precisely, there are C > 0, and spatially homogeneous distributions f+∞(v)and f−∞(v), depending continuously on fi and W , such that

(2.17) supt∈R

∥∥∥f(t, x+ vt, v) − f 0(v)∥∥∥

λ′,µ′≤ C δ;

∀ η ∈ Rd, |f±∞(η) − f 0(η)| ≤ C δ e−2πλ′|η|;

and

∀ (k, η) ∈ Zd × Rd,∣∣∣L−d f (L)(t, k, η)− f±∞(η)1k=0

∣∣∣ = O(e−2π λ′

L|t|) as t→ ±∞;

∥∥∥f(t, x+ vt, v) − f±∞(v)∥∥∥

λ′,µ′= O(e−2π λ′

L|t|) as t→ ±∞;

(2.18) ∀ r ∈ N,∥∥ρ(t, ·) − ρ∞

∥∥Cr(Td)

= O(e−2π λ′

L|t|)

as |t| → ∞;

(2.19) ∀ r ∈ N,∥∥F (t, · )‖Cr(Td) = O

(e−2π λ′

L|t|)

as |t| → ∞;

(2.20) ∀ r ∈ N, ∀σ > 0,∥∥∥⟨f(t, ·, v)

⟩−f±∞

∥∥∥Cr

σ(Rdv)

= O(e−2π λ′

L|t|)

as t→ ±∞.

In this statement Cr stands for the usual norm on r times continuously dif-ferentiable functions, and Cr

σ involves in addition moments of order σ, namely‖f‖Cr

σ= supr′≤r,v∈Rd |f (r′)(v) (1 + |v|σ)|. These results could be reformulated in

a number of alternative norms, both for the strong and for the weak topology.

2.4. Comments. Let us start with a list of remarks about Theorem 2.6.

• The decay of the force field, statement (2.19), is the experimentally measurablephenomenon which may be called Landau damping.

• Since the energy

E =1

2

∫∫ρ(x) ρ(y)W (x− y) dx dy +

∫f(x, v)

|v|22dv dx

(= potential + kinetic energy) is conserved by the nonlinear Vlasov evolution, thereis a conversion of potential energy into kinetic energy as t → ∞ (kinetic energygoes up for Coulomb interaction, goes down for Newton interaction). Similarly, theentropy

S = −∫∫

f log f = −(∫

ρ log ρ+

∫f log

f

ρ

)

(= spatial + kinetic entropy) is preserved, and there is a transfer of informationfrom spatial to kinetic variables in large time.


• Our result covers both attractive and repulsive interactions, as long as the lineardamping condition is satisfied; it covers Newton/Coulomb potential as a limit case(γ = 1 in (2.13)). The proof breaks down for γ < 1; this is a nonlinear effect, as anyγ > 0 would work for the linearized equation. The singularity of the interaction atshort scales will be the source of important technical problems.12

• Condition (2.14) could be replaced by

(2.21) |f 0(η)| ≤ C0 e−2πλ|η|,

∫f 0(v) eβ|v| dv ≤ C0.

But condition (2.14) is more general, in view of Theorem 4.20 below. For instance,f 0(v) = 1/(1+v2) in dimension d = 1 satisfies (2.14) but not (2.21); this distributionis commonly used in theoretical and numerical studies, see e.g. [38]. We shall alsoestablish slightly more precise estimates under slightly more stringent conditions onf 0, see (12.1).

• Our conditions are expressed in terms of the initial datum, which is a consid-erable improvement over [13, 40]. Still it is of interest to pursue the “scattering”program started in [13], e.g. in a hope of better understanding of the nonperturba-tive regime.

• The smallness assumption on fi − f 0 is expected, for instance in view of thework of O’Neil [73], or the numerical results of [95]. We also make the standardassumption that fi − f 0 is well localized.

• No convergence can be hoped for if the initial datum is only close to f 0 inthe weak topology: indeed there is instability in the weak topology, even around aMaxwellian [13].

• The well-posedness of the nonlinear Vlasov–Poisson equation in dimension d ≤ 3was established by Pfaffelmoser [76] and Lions–Perthame [54] in the whole space.Pfaffelmoser’s proof was adapted to the case of the torus by Batt and Rein [6]; how-ever, like Pfaffelmoser, these authors imposed a stringent assumption of uniformlybounded velocities. Building on Schaeffer’s simplification [82] of Pfaffelmoser’s ar-gument, Horst [39] proved well-posedness in the whole space assuming only inversepolynomial decay in the velocity variable. Although this has not been done explicitly,Horst’s proof can easily be adapted to the case of the torus, and covers in particularthe setting which we use in the present paper. (The adaptation of [54] seems moredelicate.) Propagation of analytic regularity is not studied in these works. In any

12In a related subject, this singularity is also the reason why the Vlasov–Poisson equation is stillfar from being established as a mean-field limit of particle dynamics (see [36] for partial resultscovering much less singular interactions).


case, our proof will provide a new perturbative existence theorem, together withregularity estimates which are considerably stronger than what is needed to provethe uniqueness. We shall not come back to these issues which are rather irrelevantfor our study: uniqueness only needs local in time regularity estimates, while all thedifficulty in the study of Landau damping consists in handling (very) large time.

• We note in passing that while blow-up is known to occur for certain solutionsof Newtonian Vlasov–Poisson in dimension 4, blow-up does not occur in this per-turbative regime, whatever the dimension. There is no contradiction since blow-upsolutions are constructed with negative energy initial data, and a nearly homoge-neous solution automatically has positive energy. (Also, blow-up solutions have beenconstructed only in the whole space, where the virial identity is available; but it isplausible, although not obvious, that blow-up is still possible in bounded geometry.)

• f(t, ·) is not close to f 0 in analytic norm as t → ∞, and does not converge toanything in the strong topology, so the conclusion cannot be improved much. Stillwe shall establish more precise quantitative results, and the limit profiles f±∞ areobtained by a constructive argument.

• Estimate (2.17) expresses the orbital “travelling stability” around f 0; it is muchstronger than the usual orbital stability in Lebesgue norms [35, 34]. An equivalentformulation is that if (Tt)t∈R stands for the nonlinear Vlasov evolution operator, and(T 0

t )t∈R for the free transport operator, then in a neighborhood of a homogeneousequilibrium satisfying the stability criterion (L), T 0

−t Tt remains uniformly close toId for all t. Note the important difference: unlike in the usual orbital stability theory,our conclusions are expressed in functional spaces involving smoothness, which arenot invariant under the free transport semigroup. This a source of difficulty (ourfunctional spaces are sensitive to the filamentation phenomenon), but it is also thereason for which this “analytic” orbital stability contains much more information,and in particular the damping of the density.

• Compared with known nonlinear stability results, and even forgetting aboutthe smoothness, estimate (2.17) is new in several respects. In the context of plasmaphysics, it is the first one to prove stability for a distribution which is not necessarily adecreasing function of |v| (“small bump on tail”); while in the context of astrophysics,it is the first one to establish stability of a homogeneous equilibrium against periodicperturbations with wavelength smaller than the Jeans length.

• While analyticity is the usual setting for Landau damping, both in mathematicaland physical studies, it is natural to ask whether this restriction can be dispended


with. (This can be done only at the price of losing the exponential decay.) In thelinear case, this is easy, as we shall recall later in Remark 3.5; but in the nonlinearsetting, leaving the analytic world is much more tricky. In Section 13, we shallpresent the first results in this direction.

With respect to the questions raised above, our analysis brings the followinganswers:

(a) Convergence of the distribution f does hold for t→ +∞; it is indeed based onphase mixing, and therefore involves very fast oscillations. In this sense it is right toconsider Landau damping as a “wild” process. But on the other hand, the spatialdensity (and therefore the force field) converges strongly and smoothly.

(b) The space average 〈f〉 does converge in large time. However the conclusions arequite different from those of quasilinear relaxation theory, since there is no need forextra randomness, and the limiting distribution is smooth, even without collisions.

(c) Landau damping is a linear phenomenon, which survives nonlinear pertur-bation thanks to the structure of the Vlasov–Poisson equation. The nonlinearitymanifests itself by the presence of echoes. Echoes were well-known to specialists ofplasma physics [52, Section 35] [1, Section 12.7], but were not identified as a possiblesource of unstability. Controlling the echoes will be a main technical difficulty; butthe fact that the response appears in this form, with an associated time-delay andlocalized in time, will in the end explain the stability of Landau damping. Thesefeatures can be expected in other equations exhibiting oscillatory behavior.

(d) The large-time limit is in general different from the limit predicted by thelinearized equation, and depends on the interaction and initial datum (more precisestatements will be given in Section 14); still the linearized equation, or higher-orderexpansions, do provide a good approximation. We shall also set up a systematicrecipe for approximating the large-time limit with arbitrarily high precision as thestrength of the perturbation becomes small. This justifies a posteriori many knowncomputations.

(e) From the point of view of dynamical systems, the nonlinear Vlasov equationexhibits a truly remarkable behavior. It is not uncommon for a Hamiltonian systemto have many, or even countably many heteroclinic orbits (there are various theoriesfor this, a popular one being the Melnikov method); but in the present case we seethat heteroclinic/homoclinic orbits13 are so numerous as to fill up a whole neigh-borhood of the equilibrium. This is possible only because of the infinite-dimensional

13Here we use these words just to designate solutions connecting two distinct/equal equilibria,without any mention of stable or unstable manifolds.


nature of the system, and the possibility to work with nonequivalent norms; sucha behavior has already been reported for other systems [48, 49], in relation withinfinite-dimensional KAM theory.

(f) As a matter of fact, nonlinear Landau damping has strong similarities withthe KAM theory. It has been known since the early days of the theory that thelinearized Vlasov equation can be reduced to an infinite system of uncoupled Volterraequations, which makes this equation completely integrable in some sense. (Morrison[64] gave a more precise meaning to this property.) To see a parallel with classicalKAM, one step of our result is to prove the preservation of the phase-mixing propertyunder nonlinear perturbation of the interaction. (Although there is no ergodicity inphase space, the mixing will imply an ergodic behavior for the spatial density.) Theanalogy is reinforced by the fact that the proof of Theorem 2.6 shares many featureswith the proof of the KAM theorem in the analytic (or Gevrey) setting. (Our proofis close to Kolmogorov’s original argument, exposed in [18].) In particular, we shallinvoke a Newton scheme to overcome a loss of “regularity” in analytic norms, only ina trickier sense than in KAM theory. If one wants to push the analogy further, onecan argue that the resonances which cause the phenomenon of small divisors in KAMtheory find an analogue in the time-resonances which cause the echo phenomenon inplasma physics. A notable difference is that in the present setting, time-resonancesarise from the nonlinearity at the level of the partial differential equation, whereassmall divisors in KAM theory arise at the level of the ordinary differential equation.Another major difference is that in the present situation there is a time-averagingwhich is not present in KAM theory.

Thus we see that three of the most famous paradoxical phenomena from twen-tieth century classical physics: Landau damping, echoes, and KAM theorem, areintimately related (only in the nonlinear variant of Landau’s linear argument!). Thisrelation, which we did not expect, is one of the main discoveries of the present paper.

2.5. Interpretation. A successful point of view adopted in this paper is that Lan-dau damping is a relaxation by smoothness and by mixing. In a way, phasemixing converts the smoothness into decay. Thus Landau damping emerges as a rareexample of a physical phenomenon in which regularity is not only crucial from themathematical point of view, but also can be “measured” by a physical experiment.

2.6. Main ingredients. Some of our ingredients are similar to those in [13]: in par-ticular, the use of Fourier transform to quantify analytic regularity and to implementphase mixing. New ingredients used in our work include


• the introduction of a time-shift parameter to keep memory of the initial time(Sections 4 and 5), thus getting uniform estimates in spite of the loss of regularityin large time. We call this the gliding regularity: it shifts in phase space from lowto high modes. Gliding regularity automatically comes with an improvement of theregularity in x, and a deterioration of the regularity in v, as time passes by.

• the use of carefully designed flexible analytic norms behaving well with respectto composition (Section 4). This requires care, because analytic norms are verysensitive to composition, contrary to, say, Sobolev norms.

• A control of the deflection of trajectories induced by the force field, to reducethe problem to homogenization of free flow (Section 5) via composition. The phys-ical meaning is the following: when a background with gliding regularity acts on(say) a plasma, the trajectories of plasma particles are asymptotic to free transporttrajectories.

• new functional inequalities of bilinear type, involving analytic functional spaces,integration in time and velocity variables, and evolution by free transport (Section6). These inequalities morally mean the following: when a plasma acts (by forcing)on a smooth background of particles, the background reacts by lending a bit of its(gliding) regularity to the plasma, uniformly in time. Eventually the plasma willexhaust itself (the force will decay). This most subtle effect, which is at the heartof Landau’s damping, will be mathematically expressed in the formalism of analyticnorms with gliding regularity.

• a new analysis of the time response associated to the Vlasov–Poisson equation(Section 7), aimed ultimately at controlling the self-induced echoes of the plasma.For any interaction less singular than Coulomb/Newton, this will be done by an-alyzing time-integral equations involving a norm of the spatial density. To treatCoulomb/Newton potential we shall refine the analysis, considering individual modesof the spatial density.

• a Newton iteration scheme, solving the nonlinear evolution problem as a suc-cession of linear ones (Section 10). Picard iteration schemes still play a role, sincethey are run at each step of the iteration process, to estimate the deflection.

It is only in the linear study of Section 3 that the length scale L will play a crucialrole, via the stability condition (L). In all the rest of the paper we shall normalizeL to 1 for simplicity.


2.7. About phase mixing. A physical mechanism transferring energy from largescales to very fine scales, asymptotically in time, is sometimes called weak turbu-lence. Phase mixing provides such a mechanism, and in a way our study shows thatthe Vlasov–Poisson equation is subject to weak turbulence. But the phase mixinginterpretation provides a more precise picture. While one often sees weak turbulenceas a “cascade” from low to high Fourier modes, the relevant picture would ratherbe a two-dimensional figure with an interplay between spatial Fourier modes andvelocity Fourier modes. More precisely, phase mixing transfers the energy from eachnonzero spatial frequency k, to large velocity frequences η, and this transfer occursat a speed proportional to k. This picture is clear from the solution of free transportin Fourier space, and is illustrated in Fig. 3. (Note the resemblance with a shearflow.) So there is transfer of energy from one variable (here x) to another (here v);homogenization in the first variable going together with filamentation in the secondone. The same mechanism may also underlie other cases of weak turbulence.

k

(kinetic modes)

initial configuration(t = 0)

(spatial modes) t = t1 t = t2 t = t3

−η

Figure 3. Schematic picture of the evolution of energy by free trans-port, or perturbation thereof; marks indicate localization of energy inphase space. The energy of the spatial mode k is concentrated in largetime around η ≃ −kt.


0

0.1

0.2

0.3

0.4 0 2

4 6x

v

Figure 4. The distribution function in phase space (position, veloc-ity) at a given time; notice how the fast oscillations in v contrast withthe slower variations in x.

Whether ultimately the high modes are damped by some “random” microscopicprocess (collisions, diffusion, . . . ) not described by the Vlasov–Poisson equationis certainly undisputed in plasma physics [52, Section 41]14, and is the object ofdebate in galactic dynamics; anyway this is a different story. Some mathematicalstatistical theories of Euler and Vlasov–Poisson equations do postulate the existenceof some small-scale coarse graining mechanism, but resulting in mixing rather thandissipation [78, 89].

3. Linear damping

In this section we establish Landau damping for the linearized Vlasov equation.Beforehand, let us recall that the free transport equation

(3.1)∂f

∂t+ v · ∇xf = 0

has a strong mixing property: any solution of (3.1) converges weakly in large timeto a spatially homogeneous distribution equal to the space-averaging of the initialdatum. Let us sketch the proof.

If f solves (3.1) in Td × Rd, with initial datum fi = f(0, · ), then f(t, x, v) =fi(x− vt, v), so the space-velocity Fourier transform of f is given by the formula

(3.2) f(t, k, η) = fi(k, η + kt).

14See [52, Problem 41]: thanks to Landau damping, collisions are expected to smooth thedistribution quite efficiently; this is a hypoelliptic problematic.


On the other hand, if f∞ is defined by

f∞(v) = 〈fi( · , v)〉 =

∫

Td

fi(x, v) dx,

then f∞(k, η) = fi(0, η) 1k=0. So, by the Riemann–Lebesgue lemma, for any fixed(k, η) we have ∣∣∣f(t, k, η) − f∞(k, η)

∣∣∣ −−−→|t|→∞

0,

which shows that f converges weakly to f∞. The convergence holds as soon as fi ismerely integrable; and by (3.2), the rate of convergence is determined by the decay

of fi(k, η) as |η| → ∞, or equivalently the smoothness in the velocity variable. Inparticular, the convergence is exponentially fast if (and only if) fi(x, v) is analyticin v.

This argument obviously works independently of the size of the box. But whenwe turn to the Vlasov equation, length scales will matter, so we shall introduce alength L > 0, and work in Td

L = Rd/(LZd). Then the length scale will appear inthe Fourier transform: see Appendix A.3. (This is the only section in this paperwhere the scale will play a nontrivial role, so in all the rest of the paper we shalltake L = 1.)

Any velocity distribution f 0 = f 0(v) defines a stationary state for the nonlin-ear Vlasov equation with interaction potential W . Then the linearization of thatequation around f 0 yields

(3.3)

∂f

∂t+ v · ∇xf − (∇W ∗ ρ) · ∇vf

0 = 0

ρ =

∫f dv.

Note that there is no force term in (3.3), due to the fact that f 0 does not dependon x. This equation describes what happens to a plasma density f which tries toforce a stationary homogeneous background f 0; equivalently, it describes the reactionexerted by the background which is acted upon. (Imagine that there is an exchangeof matter between the forcing gas and the forced gas, and that this exchange exactlycompensates the effect of the force, so that the density of the forced gas does notchange after all.)

Theorem 3.1 (Linear Landau damping). Let f 0 = f 0(v), L > 0, W : TdL → R such

that W (−z) = W (z) and ‖∇W‖L1 ≤ CW < +∞, and let fi = fi(x, v), such that

(i) Condition (L) from Subsection 2.2 holds for some constants λ, κ > 0;


(ii) ∀ η ∈ Rd, |f 0(η)| ≤ C0 e−2πλ|η| for some constant C0 > 0;

(iii) ∀ k ∈ Zd, ∀ η ∈ Rd, |f (L)i (k, η)| ≤ Ci e

−2πα|η| for some constants α > 0,Ci > 0.

Then as t → +∞ the solution f(t, ·) to the linearized Vlasov equation (3.3) withinitial datum fi converges weakly to f∞ = 〈fi〉 defined by

f∞(v) =1

Ld

∫

TdL

fi(x, v) dx;

and ρ(x) =

∫f(x, v) dv converges strongly to the constant

ρ∞ =1

Ld

∫∫

TdL×Rd

fi(x, v) dx dv.

More precisely, for any λ′ < minλ ; α,

∀ r ∈ N,∥∥ρ(t, ·) − ρ∞

∥∥Cr = O

(e−

2πλ′

L|t|)

∀ (k, η) ∈ Zd × Zd,∣∣∣f (L)(t, k, η) − f (L)

∞ (k, η)∣∣∣ = O

(e−

2πλ′

L|kt|).

Remark 3.2. Even if the initial datum is more regular than analytic, the conver-gence will in general not be better than exponential (except in some exceptionalcases [37]). See [10, pp. 414–416] for an illustration. Conversely, if the analyticitywidth α for the initial datum is smaller than the “Landau rate” λ, then the rate ofdecay will not be better than O(e−αt). See [7, 19] for a discussion of this fact, oftenoverlooked in the physical literature.

Remark 3.3. The fact that the convergence is to the average of the initial da-tum will not survive nonlinear perturbation, as shown by the counterexamples inSubsection 14.

Remark 3.4. Dimension does not play any role in the linear analysis. This can beattributed to the fact that only longitudinal waves occur, so everything happens “inthe direction of the wave vector”. Transversal waves arise in plasma physics onlywhen magnetic effects are taken into account [1, Chapter 5].

Remark 3.5. The proof can be adapted to the case when f 0 and fi are only C∞;then the convergence is not exponential, but still O(t−∞). The regularity can also befurther decreased, down to W s,1, at least for any s > 2; more precisely, if f 0 ∈W s0,1


and fi ∈W si,1 there will be damping with a rate O(t−κ) for any κ < maxs0−2 ; si.(Compare with [1, Vol. 1, p. 189].) This is independent of the regularity of theinteraction.

The proof of Theorem 3.1 relies on the following elementary estimate for Volterraequations. We use the notation of Subsection 2.2.

Lemma 3.6. Assume that (L) holds true for some constants C0, κ, λ > 0; let CW =‖W‖L1(Td

L) and let K0 be defined by (2.3). Then any solution ϕ(t, k) of

(3.4) ϕ(t, k) = a(t, k) +

∫ t

0

K0(t− τ, k)ϕ(τ, k) dτ

satisfies, for any k ∈ Zd and any λ′ < λ,

supt≥0

(|ϕ(t, k)| e2πλ′ |k|

Lt)≤[1 + C0CW C(λ, λ′, κ)

]supt≥0

(|a(t, k)| e2πλ |k|

Lt).

Here C(λ, λ′, κ) = C (1 + κ−1(1 + (λ− λ′)−2)) for some universal constant C.

Remark 3.7. It is standard to solve these Volterra equations by Laplace trans-form; but, with a view to the nonlinear setting, we shall prefer a more flexible andquantitative approach.

Proof of Lemma 3.6. If k = 0 this is obvious since K0(t, 0) = 0; so we assume k 6= 0.

Consider λ′ < λ, multiply (3.4) by e2πλ′ |k|L

t, and write

Φ(t, k) = ϕ(t, k) e2πλ′ |k|L

t, A(t, k) = a(t, k) e2πλ′ |k|L

t;

then (3.4) becomes

(3.5) Φ(t, k) = A(t, k) +

∫ t

0

K0(t− τ, k) e2πλ′ |k|L

(t−τ) Φ(τ, k) dτ.

A particular case: The proof is extremely simple if we make the stronger assump-tion ∫ +∞

0

|K0(τ, k)| e2πλ′ |k|L

τ dτ ≤ 1 − κ, κ ∈ (0, 1).

Then from (3.5),

sup0≤t≤T

|Φ(t, k)| ≤ sup0≤t≤T

|A(t, k)|

+ sup0≤t≤T

(∫ t

0

∣∣K0(t− τ, k)∣∣ e2πλ′ |k|

L(t−τ) dτ

)sup

0≤τ≤T|Φ(τ, k)|,


whence

sup0≤τ≤t

|Φ(τ, k)| ≤sup

0≤τ≤t|A(τ, k)|

1 −∫ +∞

0

|K0(τ, k)| e2πλ′ |k|L

τ dτ

≤sup

0≤τ≤t|A(τ, k)|

κ,

and therefore

supt≥0

(e2πλ′ |k|

Lt|ϕ(t, k)|

)≤(

1

κ

)supt≥0

(|a(t, k)| e2πλ′ |k|

Lt).

The general case: To treat the general case we take the Fourier transform in thetime variable, after extending K, A and Φ by 0 at negative times. (This presentationwas suggested to us by Sigal, and appears to be technically simpler than the use ofthe Laplace transform.) Denoting the Fourier transform with a hat and recalling(2.4), we have, for ξ = λ′ + iωL/|k|,

Φ(ω, k) = A(ω, k) + L(ξ, k) Φ(ω, k).

By assumption L(ξ, k) 6= 1, so

Φ(ω, k) =A(ω, k)

1 − L(ξ, k).

From there, it is traditional to apply the Fourier (or Laplace) inversion transform.Instead, we apply Plancherel’s identity to find (for each k)

‖Φ‖L2(dt) ≤‖A‖L2(dt)

κ;

and then we plug this in the equation (3.5) to get

‖Φ‖L∞(dt) ≤ ‖A‖L∞(dt) +∥∥∥K0 e2πλ′ |k|

Lt∥∥∥

L2(dt)‖Φ‖L2(dt)(3.6)

≤ ‖A‖L∞(dt) +

∥∥∥K0 e2πλ′ |k|L

t∥∥∥

L2(dt)‖A‖L2(dt)

κ.


It remains to bound the second term. On the one hand,

‖A‖L2(dt) =

(∫ ∞

0

|a(t, k)|2 e4πλ′ |k|L

t dt

)1/2

(3.7)

≤(∫ ∞

0

e−4π(λ−λ′)|k|L

t

)1/2

supt≥0

(|a(t, k)| e2πλ

|k|L

t)

=

(L

4π|k| (λ− λ′)

) 12

supt≥0

(|a(t, k)| e2πλ |k|

Lt).

On the other hand,

∥∥∥K0 e2πλ′ |k|L

t∥∥∥

L2(dt)= 4π2 |W (L)(k)| |k|

2

L2

(∫ ∞

0

e4πλ′ |k|L

t

∣∣∣∣f 0

(kt

L

)∣∣∣∣2

t2 dt

)1/2

(3.8)

= 4π2 |W (L)(k)| |k|1/2

L1/2

(∫ ∞

0

e4πλ′u |f 0(σ u)|2 u2 du

)1/2

,

where σ = k/|k| and u = |kt|/L. The estimate follows since∫∞

0e−4π(λ−λ′)u u2 du =

O((λ− λ′)−3/2). (Note that the factor |k|−1/2 in (3.7) cancels with |k|1/2 in (3.8).)

It seems that we only used properties of the function L in a strip ℜe ξ ≃ λ;but this is an illusion. Indeed, we have taken the Fourier transform of Φ withoutchecking that it belongs to (L1 + L2)(dt), so what we have established is only ana priori estimate. To convert it into a rigorous result, one can use a continuityargument after replacing λ′ by a parameter α which varies from −ǫ to λ′. (By theintegrability of K0 and Gronwall’s lemma, ϕ is obviously bounded as a function oft; so ϕ(k, t) e−ǫ|k|t/L is integrable for any ǫ > 0, and continuous as ǫ → 0.) Thenassumption (L) guarantees that our bounds are uniform in the strip 0 ≤ ℜe ξ ≤ λ′,and the proof goes through.

Proof of Theorem 3.1. Without loss of generality we assume t ≥ 0. Considering(3.3) as a perturbation of free transport, we apply Duhamel’s formula to get

(3.9) f(t, x, v) = fi(x− vt, v) +

∫ t

0

[(∇W ∗ ρ) · ∇vf

0](τ, x− v(t− τ), v

)dτ.

Integration in v yields

(3.10) ρ(t, x) =

∫

Rd

fi(x−vt, v) dv+∫ t

0

∫

Rd

[(∇W ∗ρ)·∇vf

0](τ, x−v(t−τ), v

)dv dτ.

Of course,∫ρ(t, x) dx =

∫∫fi(x, v) dx dv.


For k 6= 0, taking the Fourier transform of (3.10), we obtain

ρ(L)(t, k) =

∫

TdL

∫

Rd

fi(x− vt, v) e−2iπ kL·x dv dx

+

∫ t

0

∫

TdL

∫

Rd

[(∇W ∗ ρ) · ∇vf

0](τ, x− v(t− τ), v

)e−2iπ k

L·x dv dx dτ

=

∫

TdL

∫

Rd

fi(x, v) e−2iπ k

L·x e−2iπ k

L·vt dv dx

+

∫ t

0

∫

TdL

∫

Rd

[(∇W ∗ ρ) · ∇vf

0](τ, x, v) e−2iπ k

L·x e−2iπ k

L·v(t−τ) dv dx dτ

= f(L)i

(k,kt

L

)+

∫ t

0

(∇W ∗ ρ)b(L)(τ, k) · ∇vf 0

(k(t− τ)

L

)dτ

= f(L)i

(k,kt

L

)+

∫ t

0

(2iπ

k

LW (L)(k) ρ(L)(τ, k)

)·(

2iπk(t− τ)

Lf 0

(k(t− τ)

L

))dτ.

In conclusion, we have established the closed equation on ρ(L):

(3.11) ρ(L)(t, k) = f(L)i

(k,kt

L

)

− 4π2 W (L)(k)

∫ t

0

ρ(L)(τ, k) f 0

(k(t− τ)

L

) |k|2L2

(t− τ) dτ.

Recalling (2.3), this is the same as

ρ(L)(t, k) = f(L)i

(k,kt

L

)+

∫ t

0

K0(t− τ, k) ρ(L)(τ, k) dτ.

Without loss of generality, λ ≤ α where α appears in Theorem 3.1. By Assumption(L) and Lemma 3.6,

∣∣ρ(L)(t, k)∣∣ ≤ C0CW C(λ, λ′, κ)Ci e

−2π λ′ |k|L

t.

In particular, for k 6= 0 we have

∀ t ≥ 1, |ρ(L)(t, k)| = O(e−

2πλ′′

Lt e−

2π(λ′−λ′′)L

|k|);


so any Sobolev norm of ρ− ρ∞ converges to zero like O(e−2πλ′′

Lt), where λ′′ is arbi-

trarily close to λ′ and therefore also to λ. By Sobolev embedding, the same is truefor any Cr norm.

Next, we go back to (3.9) and take the Fourier transform in both variables x andv, to find

f (L)(t, k, η) =

∫

Td

∫

Rd

fi(x− vt, v) e−2iπ kL·x e−2iπη·v dx dv

+

∫ t

0

∫

Td

∫

Rd

(∇W ∗ ρ)(τ, x− v(t− τ)

)· ∇vf

0(v) e−2iπ kL·x e−2iπη·v dx dv dτ

=

∫

Td

∫

Rd

fi(x, v) e−2iπ k

L·x e−2iπ k

L·vt e−2iπη·v dx dv

+

∫ t

0

∫

Td

∫

Rd

(∇W ∗ ρ)(τ, x) · ∇vf0(v) e−2iπ k

L·x e−2iπ k

L·v(t−τ) e−2iπη·v dx dv dτ

= f(L)i

(k, η +

kt

L

)+

∫ t

0

∇W (L)(k) ρ(L)(τ, k) · ∇vf 0

(η +

k

L(t− τ)

)dτ.

So(3.12)

f (L)

(t, k, η − kt

L

)= f

(L)i (k, η) +

∫ t

0

∇W (L)(k) ρ(L)(τ, k) · ∇vf 0

(η − kτ

L

)dτ.

In particular, for any η ∈ Rd,

(3.13) f (L)(t, 0, η) = f(L)i (0, η);

in other words, 〈f〉 =∫f dx remains equal to 〈fi〉 for all times.


On the other hand, if k 6= 0,

∣∣∣∣f (L)

(t, k, η − kt

L

)∣∣∣∣ ≤∣∣f (L)

i (k, η)∣∣

(3.14)

+

∫ t

0

∣∣∇W (L)(k)∣∣ ∣∣ρ(L)(τ, k)

∣∣∣∣∣∣∇vf 0

(η − kτ

L

)∣∣∣∣ dτ

≤ Ci e−2πα|η|

+

∫ t

0

CW C(λ, λ′, κ)Ci e−2πλ′ |k|

Lτ

(2πC0

∣∣∣∣η −kτ

L

∣∣∣∣ e−2πλ|η− kτ

L |)dτ

≤ C

(e−2πα|η| +

∫ t

0

e−2πλ′ |k|L

τ e−2π (λ′+λ)2 |η− kτ

L | dτ),

where we have used λ′ < (λ′ + λ)/2 < λ, and C only depends on CW , Ci, λ, λ′, κ.

In the end,∫ t

0

e−2πλ′ |k|L

τ e−2π(λ′+λ)

2 |η− kτL | dτ ≤

∫ t

0

e−2πλ′|η| e−2π(λ−λ′)

2 |η− kτL | dτ

≤ L

π(λ− λ′)e−2π

“λ′−

(λ−λ′)2

”|η|.

Plugging this back in (3.14), we obtain, with λ′′ = λ′ − (λ− λ′)/2,

(3.15)

∣∣∣∣f (L)

(t, k, η − kt

L

)∣∣∣∣ ≤ C e−2πλ′′|η|.

In particular, for any fixed η and k 6= 0,∣∣f (L)(t, k, η)

∣∣ ≤ C e−2πλ′′|η+ ktL | = O

(e−2π λ′′

L|t|).

We conclude that f (L) converges pointwise, exponentially fast, to the Fourier trans-form of 〈fi〉. Since λ′ and then λ′′ can be taken as close to λ as wanted, this endsthe proof.

We close this section by proving Proposition 2.1.

Proof of Proposition 2.1. First assume (a). Since f 0 decreases exponentially fast,we can find λ, κ > 0 such that

4π2 max∣∣W (L)(k)

∣∣ sup|σ|=1

∫ ∞

0

∣∣f 0(rσ)∣∣ r e2πλr dr ≤ 1 − κ.


Performing the change of variables kt/L = rσ inside the integral, we deduce∫ ∞

0

4π2 |W (L)(k)|∣∣∣∣f 0

(kt

L

)∣∣∣∣|k|2 tL2

e2πλ |k|L

t dt ≤ 1 − κ,

and this obviously implies (L).

The choice w = 0 in (2.8) shows that Condition (b) is a particular case of (c), sowe only treat the latter assumption. The reasoning is more subtle than for case (a).

Throughout the proof we shall abbreviate W (L) into W . As a start, let us assumed = 1 and k > 0 (so k ∈ N). Then we compute: for any ω ∈ R,

∫ ∞

0

e2iπω kL

tK0(t, k) dt(3.16)

= limλ→0+

∫ ∞

0

e−2πλ kL

t e2iπω kL

tK0(t, k) dt

= −4π2 W (k) limλ→0+

∫ ∞

0

∫f 0(v) e−2iπ k

Ltv e−2πλ k

Lt e2iπω k

Lt k

2

L2t dv dt

= −4π2 W (k) limλ→0+

∫ ∞

0

∫f 0(v) e−2iπvt e−2πλt e2iπωt t dv dt.

Then by integration by parts, assuming that (f 0)′ is integrable,∫

R

f 0(v) e−2iπvt t dv =1

2iπ

∫(f 0)′(v) e−2iπvt dv.

Plugging this back in (3.16), we obtain the expression

2iπW (k) limλ→0+

∫(f 0)′(v)

∫ ∞

0

e−2π[λ+i(v−ω)]t dt dv.

Next, recall that for any λ > 0,∫ ∞

0

e−2π[λ+i(v−ω)]t dt =1

2π[λ+ i(v − ω)

] ;

indeed, both sides are holomorphic functions of z = λ + i(v − ω) in the half-planeℜe z > 0, and they coincide on the real axis z > 0, so they have to coincideeverywhere. We conclude that

(3.17)

∫ ∞

0

e2iπω kL

tK0(t, k) dt = W (k) limλ→0+

∫(f 0)′(v)

v − ω − iλdv.


The celebrated Plemelj formula states that

(3.18)1

z − i 0= p.v.

(1

z

)+ iπ δ0,

where the left-hand side should be understood as the limit, in weak sense, of1/(z − iλ) as λ → 0+. The abbreviation p.v. stands for principal value, that is,simplifying the possibly divergent part by using compensations by symmetry whenthe denominator vanishes. Formula (3.18) is proven in the Appendix A.5, where thenotion of principal value is also precisely defined. Combining (3.17) and (3.18) weend up with the identity

(3.19)

∫ ∞

0

e2iπω kL

tK0(t, k) dt = W (k)

[(p.v.

∫(f 0)′(v)

v − ωdv

)+ iπ(f 0)′(ω)

].

Since W is even, W is real-valued, so the above formula yields the decomposition ofthe limit into real and imaginary parts. The problem is to check that the real partcannot approach 1 at the same time as the imaginary part would approach 0.

As soon as (f 0)′(v) = O(1/|v|), we have∫

(f 0)′(v)/(v − ω) dv = O(1/|ω|) as|ω| → ∞, so the real part in the right-hand side of (3.19) becomes small when |ω|is large, and we can restrict to to a bounded interval |ω| ≤ Ω.

Then the imaginary part, W (k) π(f 0)′(ω), can become small only in the limitk → ∞ (but then also the real part becomes small) or if ω approaches one of thezeroes of (f 0)′. Since ω varies in a compact set, by continuity it will be sufficientto check the condition only at the zeroes of (f 0)′. In the end, we have obtained thefollowing stability criterion: for any k ∈ N,

(3.20) ∀ω ∈ R, (f 0)′(ω) = 0 =⇒ W (k)

∫(f 0)′(v)

v − ωdv 6= 1.

Now if k < 0, we can restart the computation as follows:∫ ∞

0

e2iπω |k|L

tK0(t, k) dt =

− 4π2 W (k) limλ→0+

∫ ∞

0

∫f 0(v) e−2iπ k

Ltv e−2πλ

|k|L

t e2iπω|k|L

t |k|2L2

t dv dt;

then the change of variables v → −v brings us back to the previous computationwith k replaced by |k| (except in the argument of W ) and f 0(v) replaced by f 0(−v).However, it is immediately checked that (3.20) is invariant under reversal of veloci-ties, that is, if f 0(v) is replaced by f 0(−v).


Finally let us generalize this to several dimensions. If k ∈ Zd \ 0 and ξ ∈ C, wecan use the splitting

v =k

|k| r + w, w⊥k, r =k

|k| · v

and Fubini’s theorem to rewrite

L(ξ, k) = −4π2 W (k)|k|2L2

∫ ∞

0

∫

Rd

f 0(v) e−2iπ kL

t·v t e2π |k|L

ξ∗t dv dt

= −4π2W (k)|k|2L2

∫ ∞

0

∫

R

(∫

k|k|

r+k⊥

f 0

(k

|k|r + w

)dw

)e−2iπ |k|

Lrtt e2π |k|

Lξ∗t dr dt,

where k⊥ is the hyperplane orthogonal to k. So everything is expressed in terms ofthe one-dimensional marginals of f 0. If f is a given function of v ∈ Rd, and σ is aunit vector, let us write σ⊥ for the hyperplane orthogonal to σ, and

(3.21) ∀v ∈ R fσ(v) =

∫

vσ+σ⊥

f(w) dw.

Then the computation above shows that the multidimensional stability criterionreduces to the one-dimensional criterion in each direction k/|k|, and the claim isproven.

4. Analytic norms

In this section we introduce some functional spaces of analytic functions on Rd,Td = Rd/Zd, and most importantly Td × Rd. (Changing the sidelength of the torusonly results in some changes in the constants.) Then we establish a number offunctional inequalities which will be crucial in the subsequent analysis. At the endof this section we shall reformulate the linear study in this new setting.

Throughout the whole section d is a positive integer. Working with analyticfunctions will force us to be careful with combinatorial issues, and proofs will attimes involve summation over many indices.

4.1. Single-variable analytic norms. Here “single-variable” means that the vari-able lives in either Rd or Td, but d may be greater than 1.

Among many possible families of norms for analytic functions, two will be ofparticular interest for us; they will be denoted by Cλ;p and Fλ;p. The Cλ;p normsare defined for functions on Rd or Td, while the Fλ;p norms are defined only forTd (although we could easily cook up a variant in Rd). We shall write Nd

0 for theset of d-tuples of integers (the subscript being here to insist that 0 is allowed). If


n ∈ Nd0 and λ ≥ 0 we shall write λn = λ|n|. Conventions about Fourier transform

and multidimensional differential calculus are gathered in the Appendix.

Definition 4.1 (One-variable analytic norms). For any p ∈ [1,∞] and λ ≥ 0, wedefine

(4.1) ‖f‖Cλ;p :=∑

n∈Nd0

λn

n!‖f (n)‖Lp; ‖f‖Fλ;p :=

(∑

k∈Zd

e2πλp|k| |f(k)|p)1/p

;

the latter expression standing for supk(e2πλ|k| |f(k)|) if p = ∞. We further write

(4.2) Cλ,∞ = Cλ, Fλ,1 = Fλ.

Remark 4.2. The parameter λ can be interpreted as a radius of convergence.

Remark 4.3. The norms Cλ and Fλ are of particular interest because they arealgebra norms.

We shall sometimes abbreviate ‖ · ‖Cλ;p or ‖ · ‖Fλ;p into ‖ · ‖λ;p when no confusionis possible, or when the statement works for either.

The norms in (4.1) extend to vector-valued functions in a natural way: if f is

valued in Rd or Td or Zd, define f (n) = (f(n)1 , . . . , f

(n)d ), f(k) = (f1(k), . . . , fd(k));

then the formulas in (4.1) make sense provided that we choose a norm on Rd or Td

or Zd. Which norm we choose will depend on the context; the choice will always bedone in such a way to get the duality right in the inequality |a · b| ≤ ‖a‖ ‖b‖∗. Forinstance if f is valued in Zd and g in Td, and we have to estimate f ·g, we may normZd by |k| =

∑|ki| and Td by |x| = sup |xi|.15 This will not pose any problem, and

the reader can forget about this issue; we shall just make remarks about it wheneverneeded. For the rest of this section, we shall focus on scalar-valued functions forsimplicity of exposition.

Next, we define “homogeneous” analytic seminorms by removing the zero-orderterm. We write Nd

∗ = Nd0 \ 0, Zd

∗ = Zd \ 0.

15Of course all norms are equivalent, still the choice is not innocent when the estimates areiterated infinitely many times; an advantage of the supremum norm on Rd is that it has thealgebra property.


Definition 4.4 (One-variable homogeneous analytic seminorms). For p ∈ [1,∞]and λ ≥ 0 we write

‖f‖Cλ;p =∑

n∈Nd∗

λn

n!‖f (n)‖Lp; ‖f‖Fλ;p =

∑

k∈Zd∗

e2πλp|k| |f(k)|p

1/p

.

It is interesting to note that affine functions x 7→ a · x + b can be included inCλ = Cλ;∞, even though they are unbounded; in particular ‖a · x+ b‖Cλ = λ |a|. On

the other hand, linear forms x 7−→ a ·x do not naturally belong to Fλ, because theirFourier expansion is not even summable (it decays like 1/k).

The spaces Cλ;p and Fλ;p enjoy remarkable properties, summarized in Propositions4.5, 4.8 and 4.10 below. Some of these properties are well-known, other not so.

Proposition 4.5 (Algebra property). (i) For any λ ≥ 0, and p, q, r ∈ [1,+∞] suchthat 1/p+ 1/q = 1/r, we have

‖f g‖Cλ;r ≤ ‖f‖Cλ;p ‖g‖Cλ;q.

(ii) For any λ ≥ 0, and p, q, r ∈ [1,+∞] such that 1/p+ 1/q = 1/r + 1, we have

‖f g‖Fλ;r ≤ ‖f‖Fλ;p ‖g‖Fλ;q .

(iii) As a consequence, for any λ ≥ 0, Cλ = Cλ;∞ and Fλ = Fλ;1 are normedalgebras: for either space,

‖fg‖λ ≤ ‖f‖λ ‖g‖λ.

In particular, ‖fn‖λ ≤ ‖f‖nλ for any n ∈ N0, and ‖ef‖λ ≤ e‖f‖λ.

Remark 4.6. Ultimately, property (iii) relies on the fact that L∞ and L1 are normedalgebras for the multiplication and convolution, respectively.

Remark 4.7. It follows from the Fourier inversion formula and Proposition 4.5 that‖f‖Cλ ≤ ‖f‖Fλ (and ‖f‖Cλ ≤ ‖f‖Fλ); this is a special case of Proposition 4.8 (iv)

below. The reverse inequality does not hold, because ‖f‖∞ does not control ‖f‖L1.

Analytic norms are very sensitive to composition; think that if a > 0 then‖f (a Id )‖Cλ;p = a−d/p ‖f‖Caλ;p; so we typically lose on the functional space. This isa major difference with more traditional norms used in partial differential equationstheory, such as Holder or Sobolev norms, for which composition may affect constantsbut not regularity indices. The next proposition controls the loss of regularity im-plied by composition.


Proposition 4.8 (Composition inequality). (i) For any λ > 0 and any p ∈ [1,+∞],

‖f H‖Cλ;p ≤∥∥(det∇H)−1

∥∥1/p

∞‖f‖Cν;p, ν = ‖H‖Cλ,

where H is possibly unbounded.(ii) For any λ > 0, any p ∈ [1,∞] and any a > 0,

∥∥∥f (a Id +G)∥∥∥Cλ;p

≤ a−d/p ‖f‖Caλ+ν ;p, ν = ‖G‖Cλ .

(iii) For any λ > 0,

∥∥∥f (Id +G)∥∥∥Fλ

≤ ‖f‖Fλ+ν , ν = ‖G‖Fλ.

(iv) For any λ > 0 and any a > 0,

∥∥∥f (a Id +G)∥∥∥Cλ

≤ ‖f‖Faλ+ν , ν = ‖G‖Cλ .

Remark 4.9. Inequality (iv), with C on the left and F on the right, will be mostuseful. The reverse inequality is not likely to hold, in view of Remark 4.7.

The last property of interest for us is the control of the loss of regularity involvedby differentiation.

Proposition 4.10 (Control of gradients). For any λ > λ, any p ∈ [1,+∞], we have

(4.3) ‖∇f‖Cλ;p ≤(

1

λe log(λ/λ)

)‖f‖Cλ;p;

(4.4) ‖∇f‖Fλ;p ≤(

1

2πe (λ− λ)

)‖f‖Fλ;p.

The proofs of Propositions 4.5 to 4.10 will be preparations for the more compli-cated situations considered in the sequel.


Proof of Proposition 4.5. (i) Denoting by ‖ · ‖λ;p the norm of Cλ;p, using the multi-dimensional Leibniz formula from Appendix A.2, we have

‖fg‖λ;r =∑

ℓ∈Nd0

‖(fg)(ℓ)‖Lr

λℓ

ℓ!≤∑

ℓ∈Nd0

∑

m≤ℓ

( ℓm

) ∥∥f (m)g(ℓ−m)∥∥

Lr

λℓ

ℓ!

≤∑

ℓ∈Nd0

∑

m≤ℓ

( ℓm

)‖f (m)‖Lp ‖g(ℓ−m)‖Lq

λℓ

ℓ!

=∑

ℓ

∑

m

‖f (m)‖Lp λm

m!

‖g(ℓ−m)‖Lq λℓ−m

(ℓ−m)!

= ‖f‖λ;p ‖g‖λ;q.

(ii) Denoting now by ‖ · ‖λ;p the norm of Fλ;p, and applying Young’s convolutioninequality, we get

‖fg‖λ;r =(∑

|f g(k)|r e2πλr|k|)1/r

≤(∑

k

(∑

ℓ

|f(ℓ)| |g(k − ℓ)| e2πλ|k−ℓ|e2πλ|ℓ|)r)1/r

≤(∑

k

|f(k)|p e2πλp|k−ℓ|

) 1p(∑

ℓ

|g(ℓ)|q e2πλq|ℓ|

) 1q

.

Proof of Proposition 4.8. Case (i). We use the (multi-dimensional) Faa di Brunoformula:

(f H)(n) =∑

Pnj=1 j mj=n

n!

m1! . . .mn!

(f (m1+...+mn) H

) n∏

j=1

(H(j)

j!

)mj

;

so

∥∥(f H)(n)∥∥

Lp ≤∑

Pnj=1 j mj=n

n!

m1! . . .mn!

∥∥∥f (m1+...+mn) H∥∥∥

Lp

n∏

j=1

∥∥∥∥H(j)

j!

∥∥∥∥mj

∞

;


thus

∑

n≥1

λn

n!

∥∥(f H)(n)∥∥

Lp ≤∥∥(det∇H)−1

∥∥1/p

∞

(+∞∑

k=1

‖f (k)‖Lp

∑Pn

j=1 j mj=n,Pn

j=1 mj=k

λn

m1! . . .mn!

n∏

j=1

∥∥∥∥H(j)

j!

∥∥∥∥mj

∞

)

=∥∥(det∇H)−1

∥∥1/p

∞

∑

k≥1

‖f (k)‖Lp

1

k!

∑

|ℓ|≥1

λℓ

ℓ!‖H(ℓ)‖∞

k ,

where the last step follows from the multidimensional binomial formula.

Case (ii). We decompose h(x) := f(ax+G(x)) as

h(x) =∑

n∈Nd0

(f (n))(ax)

n!G(x)n

and we apply ∇k:

∇kh(x) =∑

k1+k2=k,∈Nd0

∑

n∈Nd0

k! ak1

k1! k2!n!(∇k1+nf)(ax) (∇k2(Gn))(x).

Then we take the Lp norm, multiply by λk/k! and sum over k:

‖h‖Cλ;p ≤ |a|−d/p∑

k1,k2,n≥0

λk1+k2 |a|k1

k1! k2!n!‖∇k1+nf‖Lp

∥∥∇k2(Gn)∥∥∞

= |a|−d/p∑

k1,n≥0

λk1 |a|k1

k1!n!‖∇k1+nf‖Lp ‖Gn‖Cλ

≤ |a|−d/p∑

k1,n≥0

λk1 |a|k1

k1!n!‖∇k1+nf‖Lp ‖G‖n

Cλ

= |a|−d/p∑

m≥0

(a λ+ ‖G‖Cλ)m

m!‖∇mf‖Lp,

where Proposition 4.5 (iii) was used in the but-to-last step.


Case (iii). In this case we write, with G0 = G(0),

h(x) = f(x+G(x)) =∑

k

f(k) e2iπk·x e2iπk·G0 e2iπk·(G(x)−G0);

soh(ℓ) =

∑

k

f(k) e2iπk·G0[e2iπk·(G−G0)

]b(ℓ− k).

Then (using again Proposition 4.5)∑

ℓ

|h(ℓ)| e2πλ|ℓ| ≤∑

k

∑

ℓ

|f(k)| e2πλ|k| e2πλ|ℓ−k|∣∣∣[e2iπk·(G−G0)

]b(ℓ− k)

∣∣∣

=∑

k

|f(k)| e2πλ|k|∥∥∥e2iπk·(G−G0)

∥∥∥λ

≤∑

k

|f(k)| e2πλ|k| e‖2πk·(G−G0)‖λ

≤∑

k

|f(k)| e2πλ|k| e2π|k| ‖G−G0‖λ

= ‖f‖λ+‖G−G0‖λ= ‖f‖λ+ν , ν = ‖G‖Fλ .

Case (iv). We actually have the more precise result

(4.5) ‖f H‖Cλ ≤∑

|f(k)| e2π|k| ‖H‖Cλ .

Writing f H =∑f(k) e2iπk·H , we see that (4.5) follows from

(4.6) ‖eih‖Cλ ≤ e‖h‖Cλ .

To prove (4.6), let Pn be the polynomial in the variables Xm (m ≤ n) defined by theidentity (ef)(n) = Pn((f

(m))m≤n) ef ; this polynomial (which can be made more ex-plicit from the Faa di Bruno formula) has nonnegative coefficients, so ‖(eif )(n)‖∞ ≤Pn((‖f (m)‖)m≤n). The conclusion will follow from the identity (between formal se-ries!)

(4.7) 1 +∑

n∈Nd∗

λn

n!Pn((Xm)m≤n) = exp

∑

k∈Nd∗

λk

k!Xk

.

To prove (4.7), it is sufficient to note that the left-hand side is the expansion of eg

in powers of λ at 0, where g(λ) =∑

k∈Nd∗

λk

k!Xk.


Proof of Proposition 4.10. (a) Writing ‖ · ‖λ;p = ‖ · ‖Cλ;p, we have

‖∂if‖λ;p =∑

n

λn

n!‖∂n

x∂if‖Lp,

where ∂i = ∂/∂xi. If 1i is the d-uple of integers with 1 in position i, then (n+1i)! ≤(|n| + 1)n!, so

‖∂if‖λ;p ≤ supn

((|n| + 1)λn

λn+1

) ∑

|m|≥1

λm

m!‖∇mf‖Lp,

and the proof of (4.3) follows easily.

(b) Writing ‖ · ‖λ;p = ‖ · ‖Fλ;p, we have

‖∂if‖λ;p =

(∑

k

|ki|p |f(k)|p e2πλp |k|

)1/p

≤[supk∈Z

(|k| e2π(λ−λ) |k|

)] (∑

k∈Zd

|f(k)|p e2πλp |k|

)1/p

,

and (4.4) follows.

4.2. Analytic norms in two variables. To estimate solutions and trajectories ofkinetic equations we will work on the phase space Td

x×Rdv, and use three parameters:

λ (gliding analytic regularity); µ (analytic regularity in x); and τ (time-shift alongthe free transport semigroup). The regularity quantified by λ is said to be glidingbecause for τ = 0 this is an analytic regularity in v, but as τ grows the regularity isprogressively transferred from velocity to spatial modes, according to the evolutionby free transport. This catch is crucial to our analysis: indeed, the solution of atransport equation like free transport or Vlasov cannot be uniformly analytic16 in vas time goes by — except of course if it is spatially homogeneous. Instead, the bestwe can do is compare the solution at time τ to the solution of free transport at thesame time.

The parameters λ, µ will be nonnegative; τ will vary in R, but often be restrictedto R+, just because we shall work in positive time. When τ is not specified, thismeans τ = 0. Sometimes we shall abuse notation by writing ‖f(x, v)‖ instead of‖f‖, to stress the dependence of f on the two variables.

16By this we mean of course that some norm or seminorm quantifying the degree of analyticsmoothness in v will remain uniformly bounded.


Putting aside the time-shift for a moment, we may generalize the norms Cλ andFλ in an obvious way:

Definition 4.11 (Two-variables analytic norms). For any λ, µ ≥ 0, we define

(4.8) ‖f‖Cλ,µ =∑

m∈Nd0

∑

n∈Nd0

λn

n!

µm

m!

∥∥∥∇mx ∇n

vf∥∥∥

L∞(Tdx×Rd

v);

(4.9) ‖f‖Fλ,µ =∑

k∈Zd

∫

η∈Rd

|f(k, η)| e2πλ|η| e2πµ|k| dη.

Of course one might also introduce variants based on Lp or ℓp norms (with twoadditional parameters p, q, since one can make different choices for the space andvelocity variables).

The norm (4.9) is better adapted to the periodic nature of the problem, and is verywell suited to estimate solutions of kinetic equations (with fast decay as |v| → ∞);but in the sequel we shall also have to estimate characteristics (trajectories) whichare unbounded functions of v. We could hope to play with two different familiesof norms, but this would entail considerable technical difficulties. Instead, we shallmix the two recipes to get the following hybrid norms:

Definition 4.12 (Hybrid analytic norms). For any λ, µ ≥ 0, let

(4.10) ‖f‖Zλ,µ =∑

ℓ∈Zd

∑

n∈Nd0

λn

n!e2πµ|ℓ|

∥∥∥∇nvf(ℓ, v)

∥∥∥L∞(Rd

v).

More generally, for any p ∈ [1,∞] we define

(4.11) ‖f‖Zλ,µ;p =∑

ℓ∈Zd

∑

n∈Nd0

λn

n!e2πµ|ℓ|

∥∥∥∇nvf(ℓ, v)

∥∥∥Lp(Rd

v).

Now let us introduce the time-shift τ . We denote by (S0τ )τ≥0 the geodesic semi-

group: (S0τ )(x, v) = (x+ vτ, v). Recall that the backward free transport semigroup

is defined by (f S0τ )τ≥0, and the forward semigroup by (f S0

−τ )τ≥0.

Definition 4.13 (Time-shift pure and hybrid analytic norms).

(4.12) ‖f‖Cλ,µτ

= ‖f S0τ‖Cλ,µ =

∑

m∈Nd0

∑

n∈Nd0

λn

n!

µm

m!

∥∥∥∇mx (∇v + τ∇x)

nf∥∥∥

L∞(Tdx×Rd

v);


(4.13) ‖f‖Fλ,µτ

= ‖f S0τ‖Fλ,µ =

∑

k∈Zd

∫

η∈Rd

|f(k, η)| e2πλ|kτ+η| e2πµ|k| dη;

(4.14) ‖f‖Zλ,µτ

= ‖f S0τ‖Zλ,µ =

∑

ℓ∈Zd

∑

n∈Nd0

λn

n!e2πµ|ℓ|

∥∥∥(∇v + 2iπτℓ)nf(ℓ, v)∥∥∥

L∞(Rdv)

;

(4.15) ‖f‖Zλ,µ;pτ

=∑

ℓ∈Zd

∑

n∈Nd0

λn

n!e2πµ|ℓ|

∥∥∥(∇v + 2iπτℓ)nf(ℓ, v)∥∥∥

Lp(Rdv).

This choice of norms is one of the cornerstones of our analysis: first, because oftheir hybrid nature, they will connect well to both periodic (in x) estimates on theforce field, and uniform (in v) estimates on the “deflection maps” studied in Section5. Secondly, they are well-behaved with respect to the properties of free transport,allowing to keep track of the initial time. Thirdly, they will satisfy the algebraproperty (for p = ∞), the composition inequality and the gradient inequality (forany p ∈ [1,∞]). Before going on with the proof of these properties, we note thefollowing alternative representations.

Proposition 4.14. The norm Zλ,µ;pτ admits the alternative representations:

(4.16) ‖f‖Zλ,µ;pτ

=∑

ℓ∈Zd

∑

n∈Nd0

λn

n!e2πµ|ℓ|

∥∥∥∇nv

(f(ℓ, v) e2iπτℓ·v

)∥∥∥Lp(Rd

v);

(4.17) ‖f‖Zλ,µ;pτ

=∑

n∈Nd0

λn

n!

∥∥∥∣∣∣(∇v + τ∇x)

nf∥∥∥∣∣∣µ;p,

where

(4.18) ‖|g‖|µ;p =∑

ℓ∈Zd

e2πµ|ℓ|∥∥g(ℓ, v)

∥∥Lp(Rd

v).

4.3. Relations between functional spaces. The next propositions are easilychecked.

Proposition 4.15. With the notation from Subsection 4.2, for any τ ∈ R,(i) if f is a function only of x then

‖f‖Cλ,µτ

= ‖f‖Cλ|τ |+µ, ‖f‖Fλ,µτ

= ‖f‖Zλ,µτ

= ‖f‖Fλ|τ |+µ;

(ii) if f is a function only of v then

‖f‖Cλ,µ;pτ

= ‖f‖Zλ,µ;pτ

= ‖f‖Cλ;p, ‖f‖Fλ,µτ

= ‖f‖Fλ ;


(iii) for any function f = f(x, v), if 〈 · 〉 stands for spatial average then

‖〈f〉‖Cλ;p ≤ ‖f‖Zλ,µ;pτ

;

(iv) for any function f = f(x, v),∥∥∥∥∫

Rd

f dv

∥∥∥∥Fλ|τ |+µ

≤ ‖f‖Zλ,µ;1τ

.

Remark 4.16. Note, in Proposition 4.15 (i) and (iv), how the regularity in x isimproved by the time-shift.

Proof of Proposition 4.15. Only (iv) requires some explanations. Let ρ(x) =∫f(x, v) dv.

Then for any k ∈ Zd,

ρ(k) =

∫

Rd

f(k, v) dv;

so for any n ∈ Nd0,

(2iπtk)n ρ(k) =

∫(2iπtk)nf(k, v) dv

=

∫(∇v + 2iπtk)n f(k, v) dv.

Recalling the conventions from Appendix A.1 we deduce∑

k,n

e2πµ|k| |2πλtk|nn!

|ρ(k)| ≤∑

k,n

e2πµ|k| λn

n!

∫ ∣∣∣(∇v + 2iπtk)n f(k, v)∣∣∣ dv

= ‖f‖Zλ,µ;1t

.

Proposition 4.17. With the notation from Subsection 4.2,

λ ≤ λ′, µ ≤ µ′ =⇒ ‖f‖Zλ,µτ

≤ ‖f‖Zλ′,µ′

τ.

Moreover, for τ, τ ∈ R, and any p ∈ [1,∞],

(4.19) ‖f‖Zλ,µ;pτ

≤ ‖f‖Z

λ,µ+λ|τ−τ |;pτ

.

Remark 4.18. Note carefully that the spaces Zλ,µτ are not ordered with respect

to the parameter τ , which cannot be thought of as a regularity index. We coulddispend with this parameter if we were working in time O(1); but (4.19) is of courseof absolutely no use. This means that errors on the exponent τ should remainsomehow small, in order to be controllable by small losses on the exponent µ.


Finally we state an easy proposition which follows from the time-invariance of thefree transport equation:

Proposition 4.19. For any X ∈ C,F ,Z, and any t, τ ∈ R,

‖f S0t ‖Xλ,µ

τ= ‖f‖Xλ,µ

t+τ.

Now we shall see that the hybrid norms, and certain variants thereof, enjoy prop-erties rather similar to those of the single-variable analytic norms studied before.This will be sometimes technical, and the reader who would like to reconnect tophysical problems is advised to go directly to Subsection 4.11.

4.4. Injections. In this section we relate Zλ,µ;pτ norms to more standard norms

entirely based on Fourier space. In the next theorem we write

(4.20) ‖f‖Yλ,µτ

= ‖f‖Fλ,µ;∞τ

= supk∈Zd

supη∈Rd

e2πµ|k| e2πλ|η+kτ | |f(k, η)|.

Theorem 4.20 (Injections between analytic spaces). (i) If λ, µ ≥ 0 and τ ∈ R then

(4.21) ‖f‖Yλ,µτ

≤ ‖f‖Zλ,µ;1τ

.

(ii) If 0 < λ < λ, 0 < µ < µ ≤M , τ ∈ R, then

(4.22) ‖f‖Zλ,µτ

≤ C(d, µ)

(λ− λ)d (µ− µ)d‖f‖

Yλ,µτ.

(iii) If 0 < λ < λ ≤ Λ, 0 < µ < µ ≤ M , b ≤ β ≤ B, then there is C =C(Λ,M, b, B, d) such that

‖f‖Zλ,µ;1τ

≤ C1

minλ−λ ; µ−µ

(‖f‖

Yλ,µτ

+

max

(∫∫|f(x, v)| eβ|v| dv dx

);(∫∫

|f(x, v)| eβ|v| dv dx)2)

.

Remark 4.21. The combination of (ii) and (iii), plus elementary Lebesgue inter-polation, enables to control all norms Zλ,µ;p

τ , 1 ≤ p ≤ ∞.

Proof of Theorem 4.20. By the invariance under the action of free transport, it issufficient to do the proof for τ = 0.


By integration by parts in the Fourier transform formula, we have

f(k, η) =

∫f(k, v) e−2iπη·v dv =

∫∇m

v f(k, v)e−2iπη·v

(2iπη)mdv.

So

|f(k, η)| ≤ 1

(2π|η|)m

∫|∇m

v f(k, v)| dv;

and therefore

e2πµ|k| e2πλ|η| |f(k, η)| ≤ e2πµ|k|∑

n

(2πλ)n

n!|η|n |f(k, η)|

≤ e2πµ|k|∑

n

λn

n!

∫|∇n

v f(k, v)| dv.

This establishes (i).

Next, by differentiating the identity

f(k, v) =

∫f(k, η) e2iπη·v dη,

we get

(4.23) ∇mv f(k, v) =

∫f(k, η) (2iπη)m e2iπη·v dη.

Then we deduce (ii) by writing

∑

k,m

e2πµ|k| λm

m!‖∇m

v f(k, v)‖L∞(dv)

≤∑

k

e2πµ|k|

∫e2πλ|η||f(k, η)| dη

≤(∑

k

e−2π(µ−µ)|k|

)(∫e−2π(λ−λ)|η| dη

)(supk,η

e2πλ|η| e2πµ|k| |f(k, η)|).

The proof of (iii) is the most tricky. We start again from (4.23), but now weintegrate by parts in the η variable:

(4.24) ∇mv f(k, v) = (−1)q

∫∇q

η

[f(k, η) (2iπη)m

] e2iπη·v

(2iπv)qdv,

where q = q(v) is a multi-index to be chosen.


We split Rdv into 2d disjoint regions ∆(i1, . . . , in), where the ij are distinct indices

in 1, . . . , d:

∆(I) =v ∈ Rd; |vi| ≥ 1 ∀ i ∈ I, |vi| < 1 ∀ i /∈ I

.

If v ∈ ∆(i1, . . . , in) we apply (4.24) with the multi-index q defined by qj = 2 ifj ∈ i1, . . . , in, qj = 0 otherwise. This gives

∫

∆(i1,...,in)

|∇mv f(k, v)| dv ≤

(1

(2π)2n

∫

∆(i1,...,in)

dvi1 . . . dvin

|vi1 |2 . . . |vin|2)

supk,η

∣∣∣∇qη

[f(k, η) (2iπη)m

]∣∣∣.

Summing up all pieces and using the Leibniz formula, we get

∫|∇m

v f(k, v)| dv ≤ C(d)(1 +m2d) supk,η

sup|q|≤2d

|∇qηf(k, η)| |2πη|m−q.

At this point we apply Lemma 4.22 below with

ε =1

4min

λ− λ

λ;µ− µ

µ

,

and we get, for q ≤ 2d,

|∇qηf(k, η)| ≤ C(d)

maxn

λλ−λ

; µµ−µ

o

K(b, B) e−2π λ+λ2

|η|

(sup

ηe2πλ|η||f(k, η)|

)1−ε

max

(supℓ,η

βℓ ‖∇ℓηf‖∞ℓ!

)ε

;

(supℓ,η

βℓ ‖∇ℓηf‖∞ℓ!

)2ε .

Of course,

βℓ |∇ℓηf(k, η)|ℓ!

≤ (2πβ)ℓ

∫

Rd

|f(k, v)| |v|ℓ

ℓ!dv

≤∫

Rd

|f(x, v)| (2πβ)ℓ |v|ℓℓ!

dv ≤∫

Rd

|f(x, v)| e2πβ|v| dv.


So, all in all,

∑

k,m

e2πµ|k| λm

m!

∫|∇m

v f(k, v)| dv

≤∑

|q|≤2d

C(d,Λ,M, b, B)1

minλ−λ ; µ−µ

supη∈Rd

(e−2π λ+λ

2|η|∑

m

λm(1 +m)2d |2πη|m−q

m!

) (∑

k

e−2π(µ(1−ε)−µ)|k|

)

(supk,η

e2πµ|k| e2πλ|η| |f(k, η)|)1−ε

max

(∫

Rd

|f(x, v)| eβ|v| dv

)ε

;

(∫

Rd

|f(x, v)| eβ|v| dv

)2ε.

Since∑

m

λm(1 +m)2d |2πη|m−q

m!≤ C(q,Λ) e2π λ+λ

2|η|

and ∑

k

e−2π(µ(1−ε)−µ)|k| ≤∑

k

e−π(µ−µ)|k| ≤ C/(µ− µ)d,

we easily end up with the desired result.

Lemma 4.22. Let f : Rd → C, and let α > 0, A ≥ 1, q ∈ Nd0. Let β such that

0 < b ≤ β ≤ B. If |f(x)| ≤ Ae−α|x| for all x, then for any ε ∈ (0, 1/4) one has

(4.25) |∇qf(x)| ≤ C(q, d)1ε K(b, B)A1−ε e−(1−2ε)α|x|

supr∈Nd

0

max

(βr ‖∇rf‖∞

r!

)ε

;

(βr ‖∇rf‖∞

r!

)2ε.

Remark 4.23. One may conjecture that the optimal constant in the right-hand sideof (4.25) is in fact polynomial in 1/ε; if this conjecture holds true, then the constantsin Theorem 4.20 (iii) can be improved accordingly. Mironescu communicated to usa derivation of polynomial bounds for the optimal constant in the related inequality

‖f (k)‖L∞(R) ≤ C(k) ‖f‖1/(k+2)

L1(R) ‖f (k+1)‖(k+1)/(k+2)L∞(R) ,

based on a real interpolation method.


Proof of Lemma 4.22. Let us first see f as a function of x1, and treat x′ = (x2, . . . , xd)as a parameter. Thus the assumption is |f(x1, x

′)| ≤ (Ae−α|x′|) e−α|x1|. By a moreor less standard interpolation inequality [22, Lemma A.1],

(4.26) |∂1f(x1, x′)| ≤ 2

√Ae−α|x′|

√e−α|x1| ‖∂2

1f(x1, x′)‖

12∞ = 2

√Ae−α|x|

√‖∂2

1f‖∞.Let Cq1,r1 be the optimal constant (not smaller than 1) such that

(4.27) |∂q1

1 f(x1, x′)| ≤ Cq1,r1 (Ae−α|x|)

1−q1r1 ‖∂r1

r f(x1, x′)‖

q1r1∞ .

By iterating (4.26), we find Cq1,r1 ≤ 2√Cq1−1,r1 Cq1+1,r1 . It follows by induction that

Cq,r ≤ 2q(r−q).

Next, using (4.27) and interpolating according to the second variable x2 as in(4.26), we get

|∂q22 ∂

q11 f(x)| ≤ Cq2,r2

(Cq1,r1 (Ae−α|x|)

1−q1r1 ‖∂r1

1 f‖q1r1∞

)1−q2r2 ‖∂r2

2 ∂q11 f‖

q2r2∞

≤ Cq1,r1 Cq2,r2 (Ae−α|x|)(1−

q1r1

)(1−q2r2

) ‖∂r11 f‖

q1r1

(1−q2r2

)

∞ ‖∂r22 ∂

q11 f‖

q2r2∞ .

We repeat this until we get

(4.28) |∇qf(x)| ≤ (Cq1,r1 . . . Cqd,rd) (Ae−α|x|)

(1−q1r1

)...(1−qdrd

)

‖∂r11 f‖

q1r1

(1−q2r2

)...(1−qdrd

)

∞ ‖∂q1

1 ∂r22 f‖

q2r2

(1−q3r3

)...(1−qdrd

). . . ‖∂q1

1 ∂q2

2 . . . ∂qd−1

d−1 ∂rdd f‖

qdrd .

Choose ri (1 ≤ i ≤ d) in such a way that

ε

d≤ qiri

≤ 2ε

d;

this is always possible for ε < d/4. Then Cqi,ri≤ (2dq2

i )1/ε, and (4.28) implies

|∇qf(x)| ≤ (2d|q|2)1/ε (Ae−α|x|)1−ε maxs≤r+q

‖∇sf‖ε

∞ ; ‖∇sf‖2ε∞

.

Then, since 2(r+ q)ε ≤ 3dq we have, by a crude application of Stirling’s formula (inquantitative form), for s ≤ r + q,

‖∇sf‖ε∞ ≤

(βs ‖∇sf‖∞

s!

)ε (s!

βs

)ε

≤(

supn

βn ‖∇nf‖∞n!

)ε

C(β, q, d) ε−3dq,


and the result follows easily.

4.5. Algebra property in two variables. In this section we only consider thenorms Zλ,µ;p

τ ; but similar results would hold true for the two-variables C and Fspaces, and could be proven with the same method as those used for the one-variablespaces Fλ and Cλ respectively (note that the Leibniz formula still applies because∇x and (∇v + τ∇x) commute).

Proposition 4.24. (i) For any λ, µ ≥ 0, τ ∈ R and p, q, r ∈ [1,+∞] such that1/p+ 1/q = 1/r, we have

‖f g‖Zλ,µ;rτ

≤ ‖f‖Zλ,µ;pτ

‖g‖Zλ,µ;qτ

.

(ii) As a consequence, Zλ,µτ = Zλ,µ;∞

τ is a normed algebra:

‖fg‖Zλ,µτ

≤ ‖f‖Zλ,µτ

‖g‖Zλ,µτ.

In particular, ‖fn‖Zλ,µτ

≤ ‖f‖nZλ,µ

τfor any n ∈ N0, and ‖ef‖Zλ,µ

τ≤ e

‖f‖Z

λ,µτ .

Proof of Proposition 4.24. First we note that (with the notation (4.18)) ‖| · ‖|µ;r sat-isfies the “(p, q, r) property”: whenever p, q, r ∈ [1,+∞] satisfy 1/p+ 1/q = 1/r, wehave

‖|fg‖|µ;r =∑

ℓ∈Zd

e2πµ|ℓ| ‖f g(ℓ, · )‖Lr(Rdv)

=∑

ℓ∈Zd

e2πµ|ℓ|

∥∥∥∥∥∑

k

f(k, · ) g(ℓ− k, · )∥∥∥∥∥

Lr(Rdv)

≤∑

ℓ∈Zd

∑

k∈Zd

e2πµ|k| e2πµ|ℓ−k| ‖f(k, · )‖Lp(Rdv) ‖g(ℓ− k, · )‖Lq(Rd

v)

= ‖|f‖|µ;p ‖|g‖|µ;q.


Next, we write

‖|fg‖|Zλ,µ;rτ

=∑

n∈Nd0

λn

n!‖|(∇v + τ∇x)

n(fg)‖|µ;r

=∑

n

λn

n!

∥∥∥∣∣∣∑

m≤n

( nm

)(∇v + τ∇x)

mf (∇v + τ∇x)n−mg

∥∥∥∣∣∣µ;r

≤∑

n

λn

n!

∑

m≤n

( nm

)‖|(∇v + τ∇x)

mf‖|µ;p ‖|(∇v + τ∇x)n−mg‖|µ;q

=

(∑

m

λm

m!‖|(∇v + τ∇x)

mf‖|µ;p

) (∑

ℓ

λℓ

ℓ!‖|(∇v + τ∇x)

ℓf‖|µ;q

)

= ‖|f‖|Zλ,µ;pτ

‖|g‖|Zλ,µ;qτ

.

(We could also reduce to τ = 0 by means of Proposition 4.19.)

4.6. Composition inequality.

Proposition 4.25 (Composition inequality in two variables). For any λ, µ ≥ 0 andany p ∈ [1,∞], τ ∈ R, σ ∈ R, a ∈ R \ 0, b ∈ R,

(4.29)∥∥∥f(x+ bv +X(x, v), av + V (x, v)

)∥∥∥Zλ,µ;p

τ

≤ |a|−d/p ‖f‖Zα,β;pσ

,

where

(4.30) α = λ|a| + ‖V ‖Zλ,µτ, β = µ+ λ |b+ τ − aσ| + ‖X − σV ‖Zλ,µ

τ.

Remark 4.26. The norms in (4.30) for X and V have to be based on L∞, not justany Lp. Also note: the fact that the second argument of f has the form av+V (andnot av + cx+ V ) is related to Remark 4.7.

Proof of Proposition 4.25. The proof is a combination of the arguments in Proposi-tion 4.8. In a first step, we do it for the case τ = σ = 0, and we write ‖ · ‖λ,µ;p =‖ · ‖Zλ,µ;p

0.


From the expansion f(x, v) =∑f(k, v) e2iπk·x we deduce

h(x, v) := f(x+ bv +X(x, v), av + V (x, v)

)

=∑

k

f(k, av + V ) e2iπk·(x+bv+X)

=∑

k

∑

m

∇mv f(k, av) · V

m

m!e2iπk·x e2iπk·bv e2iπk·X .

Taking the Fourier transform in x, we see that for any ℓ ∈ Zd,

h(ℓ, v) =∑

k

∑

m

∇mv f(k, av) e2iπk·bv

∑

j

(V m)b(j)

m!(e2iπk·X)b(ℓ− k − j).

Differentiating n times via the Leibniz formula (here applied to a product of fourfunctions), we get

∇nv h(ℓ, v) =

∑

k,m,j

∑

n1+n2+n3+n4=n

n! an1

n1!n2!n3!n4!∇m+n1

v f(k, av)

∇n2v (V m)b(j)

m!∇n3

v

(e2iπk·X

)b(ℓ− k − j, v) (2iπbk)n4 e2iπk·bv.


Multiplying by λn e2πµ|ℓ|/n! and summing over n and ℓ, taking Lp norms and using‖fg‖Lp ≤ ‖f‖Lp‖g‖L∞, we finally obtain

‖h‖λ,µ ≤ |a|−d/p∑

k,j,ℓ∈Zd0; m,n, n1+n2+n3+n4=n≥0

λn e2πµ|ℓ||a|n1

n1!n2!n3!n4!

∥∥∇m+n1v f(k, ·)

∥∥Lp

∥∥∥∥∇n2

v (V m)b(j)

m!

∥∥∥∥∞

∥∥∥∇n3v (e2iπk·X)b(ℓ− k − j)

∥∥∥∞

(2π|b| |k|)n4

= |a|−d/p∑

k,j,ℓ∈Zd0, m,n1,n2,n3,n4≥0

λn1+n2+n3+n4 e2πµ|k| e2πµ|j| e2πµ|ℓ−k−j| |a|n1

n1!n2!n3!n4!‖∇m+n1

v f(k, ·)‖Lp

∥∥∥∥∇n2

v (V m)b(j)

m!

∥∥∥∥∞

∥∥∥∇n3v (e2iπk·X)b(ℓ− k − j)

∥∥∥∞

(2π|b| |k|)n4

≤ |a|−d/p∑

k,n1,m

λn1|a|n1

n1!

∥∥∇n1+mv f(k, ·)

∥∥Lp e

2πµ|k|

(1

m!

∑

n2,j

λn2

n2!e2πµ|j| ‖∇n2

v (V m)b(j)‖∞)

(∑

n3,h

λn3

n3!e2πµ|h|

∥∥∇n3v (e2iπk·X)b(h)

∥∥∞

) (∑

n4

(2πλ|b||k|)n4

n4!

)

= |a|−d/p∑

k,p,m

(λ|a|)n1

n1!e2πµ|k|

∥∥∇n1+mv f(k, ·)‖Lp

‖V m‖λ,µ

m!

∥∥e2iπk·X‖λ,µ e2πλ|b||k|

≤ |a|−d/p∑

k,n1,m

(λ|a|)n1

n1!e2π(µ+λ|b|)|k|

∥∥∇n1+mv f(k, ·)

∥∥Lp

‖V ‖mλ,µ

m!e2π|k| ‖X‖λ,µ

= |a|−d/p∑

k,n

1

n!

(λ|a| + ‖V ‖λ,µ

)n ∥∥∇nv f(k, ·)

∥∥Lp e

2π|k|(µ+λ|b|+‖X‖λ,µ)

= |a|−d/p ‖f‖λ|a|+‖V ‖λ,µ, µ+λ|b|+‖X‖λ,µ.


Now we generalize this to arbitrary values of σ and τ : by Proposition 4.19,∥∥∥f(x+ bv +X(x, v), av + V (x, v)

)∥∥∥Zλ,µ;p

τ

=∥∥∥f(x+ v(b+ τ) +X(x+ vτ, v), av + V (x+ vτ, v)

)∥∥∥Zλ,µ;p

=∥∥∥f S0

σ S0−σ

(x+ v(b+ τ) +X(x+ vτ, v), av + V (x+ vτ, v)

)∥∥∥Zλ,µ;p

=∥∥∥(f S0

σ)(x+ v(b+ τ − aσ) + (X − σV )(x+ vτ, v), av + V (x+ vτ, v)

)∥∥∥Zλ,µ;p

=∥∥∥(f S0

σ)(x+ v(b+ τ − aσ) + Y (x, v), av +W (x, v)

)∥∥∥Zλ,µ;p

,

where

W (x, v) = V S0τ (x, v), Y (x, v) = (X − σV ) S0

τ (x, v).

Applying the result for τ = 0, we deduce that the norm of h(x, v) = f(x + bv +X(x, v), av + V (x, v)) in Zλ,µ

τ is bounded by

‖f S0σ‖Zα,β;p = ‖f‖Zα,β;p

σ,

where

α = λ|a| + ‖V S0τ‖Zλ,µ = |a|λ+ ‖V ‖Zλ,µ

τ,

and

β = µ+ λ|b+ τ − aσ| + ‖(X − σV ) S0τ‖Zλ,µ = µ+ λ|b+ τ − aσ| + ‖X − σV ‖Zλ,µ

τ.

This establishes the desired bound.

4.7. Gradient inequality. In the next proposition we shall write

(4.31) ‖f‖Zλ,µτ

=∑

ℓ∈Zd\0

∑

n∈Nd0

λn

n!e2πµ|ℓ|

∥∥∥(∇v + 2iπτℓ)nf(ℓ, v)∥∥∥

L∞(Rdv).

This is again a homogeneous (in the x variable) seminorm.

Proposition 4.27. For λ > λ ≥ 0, µ > µ ≥ 0, we have the functional inequalities

‖∇xf‖Zλ,µ;pτ

≤ C(d)

(µ− µ)‖f‖Zλ,µ;p

τ;

∥∥(∇v + τ∇x)f∥∥Zλ,µ;p

τ≤ C(d)

λ log(λ/λ)‖f‖

Zλ,µ;pτ

.


In particular, for τ ≥ 0 we have

‖∇vf‖Zλ,µ;pτ

≤ C(d)

[(1

λ log(λ/λ)

)‖f‖

Zλ,µ;pτ

+

(τ

(µ− µ)

)‖f‖

Zλ,µ;pτ

].

The proof is similar to the proof of Proposition 4.10; the constant C(d) arises inthe choice of norm on Rd. As a consequence, if 1 < λ/λ ≤ 2, we have e.g. the bound

‖∇f‖Zλ,µ;pτ

≤ C(d)

(1

λ− λ+

1 + τ

µ− µ

)‖f‖

Zλ,µ;pτ

.

4.8. Inversion. From the composition inequality follows an inversion estimate.

Proposition 4.28 (Inversion inequality). (i) Let λ, µ ≥ 0, τ ∈ R, and F : Td×Rd →Td × Rd. Then there is ε = ε(d) such that if F satisfies

‖∇(F − Id )‖Zλ′,µ′

τ≤ ε(d),

whereλ′ = λ+ 2‖F − Id ‖Zλ,µ

τ, µ′ = µ+ 2(1 + |τ |) ‖F − Id ‖Zλ,µ

τ,

then F is invertible and

(4.32) ‖F−1 − Id ‖Zλ,µτ

≤ 2 ‖F − Id ‖Zλ,µτ.

(ii) More generally, if F and G are functions Td × Rd → Td × Rd such that

(4.33) ‖∇(F − Id )‖Zλ′,µ′

τ≤ ε(d),

whereλ′ = λ+ 2‖F −G‖Zλ,µ

τ, µ′ = µ+ 2(1 + |τ |) ‖F −G‖Zλ,µ

τ,

then F is invertible and

(4.34) ‖F−1 G− Id ‖Zλ,µτ

≤ 2 ‖F −G‖Zλ,µτ.

Remark 4.29. The conditions become very stringent as τ becomes large: basically,F − Id (or F −G in case (ii)) should be of order o(1/τ) for Proposition 4.28 to beapplicable.

Remark 4.30. By Proposition 4.27, a sufficient condition for (4.33) to hold is thatthere be λ′′, µ′′ such that λ ≤ λ′′ ≤ 2λ, µ ≤ µ′′, and

‖F − Id ‖Zλ′′,µ′′

τ≤ ε′(d)

1 + τminλ′′ − λ′ ; µ′′ − µ′.

However, this condition is in practice hard to fulfill.


Proof of Proposition 4.28. We prove only (ii), of which (i) is a particular case. Letf = F − Id , h = F−1 G− Id , g = G− Id , so that Id + g = (Id + f) (Id + h), orequivalently

h = g − f (Id + h).

So h is a fixed point of

Φ : Z 7−→ g − f (Id + Z).

Note that Φ(0) = g − f . If Φ is (1/2)-Lipschitz on the ball B(0, 2‖f − g‖) in Zλ,µτ ,

then (4.34) will follow by fixed point iteration as in Theorem A.2.

So let Z, Z be given with

‖Z‖Zλ,µτ, ‖Z‖Zλ,µ

τ≤ 2‖f − g‖Zλ,µ

τ.

We have

Φ(Z) − Φ(Z) = f(Id + Z) − f(Id + Z)

=

(∫ 1

0

∇f(Id + (1 − θ)Z + θZ

)dθ

)· (Z − Z).

By Proposition 4.24,

‖Φ(Z) − Φ(Z)‖Zλ,µτ

≤(∫ 1

0

∥∥∥∇f(Id + (1 − θ)Z + θZ

)∥∥∥Zλ,µ

τ

dθ

)‖Z − Z‖Zλ,µ

τ.

For any θ ∈ [0, 1], by Proposition 4.25,∥∥∥∇f

(Id + (1 − θ)Z + θZ

)∥∥∥Zλ,µ

τ

≤ ‖∇f‖Z

bλ,bµτ,

where

λ = λ+ max‖Z‖ ; ‖Z‖ ≤ λ+ 2‖f − g‖Zλ,µτ

and (writing Z = (Zx, Zv), Z = (Zx, Zv))

µ = µ+ max‖Zx − τZv‖ ; ‖Zx − τZv‖

≤ µ+ 2(1 + |τ |) ‖f − g‖Zλ,µ

τ.

If F and G satisfy the assumptions of Proposition 4.28, we deduce that

‖Φ‖Lip(B(0,2)) ≤ C(d) ε(d),

and this is bounded above by 1/2 if ε(d) is small enough.


4.9. Sobolev corrections. We shall need to quantify Sobolev regularity correctionsto the analytic regularity, in the x variable.

Definition 4.31 (Hybrid analytic norms with Sobolev corrections). For λ, µ, γ ≥ 0,τ ∈ R, p ∈ [1,∞], we define

‖f‖Z

λ,(µ,γ);pτ

=∑

ℓ∈Zd

∑

n∈Nd0

λn

n!e2πµ|ℓ| (1 + |ℓ|)γ

∥∥(∇v + 2iπτℓ)n f(ℓ, v)∥∥

Lp(Rdv)

;

‖f‖Fλ,γ =∑

k∈Zd

e2πλ|k| (1 + |k|)γ |f(k)|.

Proposition 4.32. Let λ, µ, γ ≥ 0, τ ∈ R and p ∈ [1,+∞]. We have the followingfunctional inequalities:

(i) ‖f‖Z

λ,(µ,γ);pt+τ

= ‖f S0t ‖Zλ,(µ,γ);p

τ;

(ii) 1/p+ 1/q = 1/r =⇒ ‖fg‖Z

λ,(µ,γ);rτ

≤ ‖f‖Z

λ,(µ,γ);pτ

‖g‖Z

λ,(µ,γ);qτ

and therefore in

particular Zλ,(µ,γ)τ = Zλ,(µ,γ);∞

τ is a normed algebra;

(iii) If f depends only on x then ‖f‖Z

λ,(µ,γ)τ

= ‖f‖Fλ|τ |+µ,γ ;

(iv) ‖f‖Z

λ,(µ,γ);pτ

≤ ‖f‖Z

λ,(µ+λ|τ−τ|,γ);pτ

;

(v) for any σ ∈ R, a ∈ R \ 0, b ∈ R, p ∈ [1,∞],∥∥∥f(x+ bv +X(x, v), av + V (x, v)

)∥∥∥Z

λ,(µ,γ);pτ

≤ |a|−d/p ‖f‖Z

α,(β,γ);pσ

,

where α = λ|a| + ‖V ‖Z

λ,(µ,γ)τ

and β = µ+ λ|bτ − aσ| + ‖X − σV ‖Z

λ,(µ,γ)τ

.

(vi) Gradient inequality:

‖∇xf‖Zλ,(µ,γ);pτ

≤ C(d)

µ− µ‖f‖

Zλ,(µ,γ);pτ

,

‖∇f‖Z

λ,(µ,γ);pτ

≤ C(d)

(1

λ− λ+

1 + τ

µ− µ

)‖f‖

Zλ,(µ,γ);pτ

.

(vii) Inversion: If F and G are functions Td × Rd → Td × Rd such that

‖∇(F − Id )‖Z

λ′,(µ′,γ)τ

≤ ε(d),

where

λ′ = λ+ 2‖F −G‖Z

λ,(µ,γ)τ

, µ′ = µ+ 2(1 + |τ |) ‖F −G‖Z

λ,(µ,γ)τ

,


then

(4.35) ‖F−1 G− Id ‖Z

λ,(µ,γ)τ

≤ 2 ‖F −G‖Z

λ,(µ,γ)τ

.

Proof of Proposition 4.32. The proofs are the same as for the “plain” hybrid norms;the only notable point is that for the proof of (ii) we use, in addition to e2πλ|k| ≤e2πλ|k−ℓ| e2πλ|ℓ|, the inequality

(1 + |k|)γ ≤ (1 + |k − ℓ|)γ (1 + |ℓ|)γ.

Remark 4.33. Of course, some of the estimates in Proposition 4.32 can be “im-proved” by taking advantage of γ; e.g. for γ ≥ 1 we have

‖∇xf‖Zλ,µ;pτ

≤ C(d) ‖f‖Z

λ,(µ,γ);pτ

.

4.10. Individual mode estimates. To handle very singular cases, we shall at timesneed to estimate Fourier modes individually, rather than full norms. If f = f(x, v),we write

(4.36) (Pkf)(x, v) = f(k, v) e2iπk·x.

In particular the following estimates will be useful.

Proposition 4.34. For any λ, µ ≥ 0, τ ∈ R, Lebesgue exponents 1/r = 1/p + 1/qand k ∈ Zd, we have the estimate

‖Pk(fg)‖Zλ,µ;rτ

≤∑

ℓ∈Zd

‖Pℓf‖Zλ,µ;pτ

‖Pk−ℓg‖Zλ,µ;qτ

.

Proposition 4.35. For any λ > 0, µ ≥ µ ≥ 0, τ ∈ R, p ∈ [1,∞] and k ∈ Zd, wehave the estimate∥∥∥Pk

[f(x+X(x, v), v

)]∥∥∥Zλ,µ;p

τ

≤∑

ℓ∈Zd

e−2π(µ−µ)|k−ℓ| ‖Pℓf‖Zλ,ν;pτ

, ν = µ+ ‖X‖Zλ,µτ.

These estimates also have variants with Sobolev corrections. Note that whenµ = µ, Proposition 4.35 is a direct consequence of Proposition 4.25 with V = 0,b = 0 and a = 1:∥∥∥Pk

[f(x+X(x, v), v

)]∥∥∥Zλ,µ;p

τ

≤∥∥∥f(x+X(x, v), v

)∥∥∥Zλ,µ;p

τ

≤ ‖f‖Zλ,ν;pτ

, ν = µ+‖X‖Zλ,µτ.

Proof of Propositions 4.34 and 4.35. The proof of Proposition 4.34 is quite similarto the proof of Proposition 4.24 (It is no restriction to choose τ = 0 because Pk

commutes with the free transport semigroup.) Proposition 4.35 needs a few words


of explanation. As in the proof of Proposition 4.25 we let h(x, v) = f(x+X(x, v), v),and readily obtain

‖Pkh‖Zλ,µ;pτ

=∑

n∈Nd0

λn e2πµ|k|

n!‖∇n

v h(k, v)‖Lp(dv)

≤∑

n≥0

∑

ℓ∈Zd

λn e2πµ|k|

n!‖∇n

v f(ℓ, v)‖Lp(dv)

(∑

m≥0

λm

m!

∥∥∥∇mv

(e2iπℓ X

)b(k − ℓ, v)

∥∥∥L∞(dv)

).

At this stage we write

e2πµ|k| ≤ e2πµ|ℓ| e−2π(µ−µ)|k−ℓ| e2πµ|k−ℓ|,

and use the crude bound

∀ ℓ ∈ Zd, e2πµ|k−ℓ|∥∥∥∇m

v

(e2iπℓ X

)b(k − ℓ, v)

∥∥∥L∞(dv)

≤∑

j∈Zd

e2πµ|j|∥∥∥∇m

v

(e2iπℓ X

)b(j, v)

∥∥∥L∞(dv)

.

The rest of the proof is as in Proposition 4.25.

4.11. Measuring solutions of kinetic equations in large time. As we alreadydiscussed, even for the simplest kinetic equation, namely free transport, we cannothope to have uniform in time regularity estimates in the velocity variable: rather,because of filamentation, we may have ‖∇vf(t, ·)‖ = O(t), ‖∇2

vf(t, ·)‖ = O(t2), etc.For analytic norms we may at best hope for an exponential growth.

But the invariance of the “gliding” norms Zλ,µτ under free transport (Proposition

4.19) makes it possible to look for uniform estimates such as

(4.37) ‖f(τ, ·)‖Zλ,µτ

= O(1) as τ → +∞.

Of course, by Proposition 4.27, (4.37) implies

(4.38) ‖∇vf(τ, ·)‖Zλ′,µ′

τ= O(τ), λ′ < λ, µ′ < µ,

and nothing better as far as the asymptotic behavior of ∇vf is concerned; but (4.37)is much more precise than (4.38). For instance it implies ‖(∇v +τ∇x)f(τ, ·)‖

Zλ′,µ′τ

=

O(1) for λ′ < λ, µ′ < µ.Another way to get rid of filamentation is to average over the spatial variable x,

a common sense procedure which has already been used in physics [52, Section 49].Think that, if f evolves according to free transport, or even according to the lin-earized Vlasov equation (3.3), then its space-average

(4.39) 〈f〉(τ, v) :=

∫

Td

f(τ, x, v) dx


is time-invariant. (We used this infinite number of conservation laws to determinethe long-time behavior in Theorem 3.1.)

The bound (4.37) easily implies a bound on the space average: indeed,

(4.40) ‖〈f〉(τ, ·)‖Cλ = ‖〈f〉(τ, ·)‖Zλ,µτ

≤ ‖f(τ, ·)‖Zλ,µτ

= O(1) as τ → ∞;

and in particular, for λ′ < λ,

(4.41) ‖〈∇vf〉(τ, ·)‖Cλ′ = O(1) as τ → ∞.

Again, (4.37) contains a lot more information than (4.41).

Remark 4.36. The idea to estimate solutions of a nonlinear equation by comparisonto some unperturbed (reversible) linear dynamics is already present in the definitionof Bourgain spaces Xs,b [12]. The analogy stops here, since time is a dummy variable

in Xs,b spaces, while in Zλ,µt spaces it is frozen and appears as a parameter, on which

we shall play later.

4.12. Linear damping revisited. As a simple illustration of the functional analy-sis introduced in this section, let us recast the linear damping (Theorem 3.1) in thislanguage. This will be the first step for the study of the nonlinear damping. Forsimplicity we set L = 1.

Theorem 4.37 (Linear Landau damping again). Let f 0 = f 0(v), W : Td → R suchthat ‖∇W‖L1 ≤ CW , and fi(x, v) such that

(i) Condition (L) from Subsection 2.2 holds for some constants C0, λ, κ > 0;

(ii) ‖f 0‖Cλ;1 ≤ C0;

(iii) ‖fi‖Zλ,µ;1 ≤ δ for some µ > 0, δ > 0;

Then for any λ′ < λ and µ′ < µ, the solution of the linearized Vlasov equation (3.3)satisfies

(4.42) supt∈R

∥∥f(t, · )∥∥Zλ′,µ′;1

t≤ C δ,

for some constant C = C(d, CW , C0, λ, λ′, µ, µ′, κ). In particular, ρ =

∫f dv satisfies

(4.43) supt∈R

∥∥ρ(t, · )∥∥Fλ′|t|+µ′ ≤ C δ.

As a consequence, as |t| → ∞, ρ converges strongly to ρ∞ =∫∫

fi(x, v) dx dv, and

f converges weakly to 〈fi〉 =∫fi dx, at rate O(e−λ′′|t|) for any λ′′ < λ′.


If moreover ‖f 0‖Cλ;p ≤ C0 and ‖fi‖Zλ,µ;p ≤ δ for all p in some interval [1, p], then(4.42) can be reinforced into

(4.44) supt∈R

∥∥f(t, · )∥∥Zλ′,µ′;p

t≤ C δ, 1 ≤ p ≤ p.

Remark 4.38. The notions of weak and strong convergence are the same as thosein Theorem 3.1. With respect to that statement, we have added an extra analyticityassumption in the x variable; in this linear context this is an overkill (as the proofwill show), but later in the nonlinear context this will be important.

Proof of Theorem 4.37. Without loss of generality we restrict our attention to t ≥ 0.Although (4.43) follows from (4.42) by Proposition 4.15, we shall establish (4.43)first, and deduce (4.42) thanks to the equation. We shall write C for various con-stants depending only on the parameters in the statement of the theorem.

As in the proof of Theorem 3.1, we have

ρ(t, k) = fi(k, kt) +

∫ t

0

K0(t− τ, k) ρ(τ, k) dτ

for any t ≥ 0, k ∈ Zd. By Lemma 3.6, for any λ′ < λ, µ′ < µ,

supt≥0

(∑

k

|ρ(t, k)| e2π(λ′t+µ′)|k|

)

≤ C(λ, λ′, κ)

(∑

k

e−2π(µ−µ′)|k|

)supt≥0

supk∈Zd

∣∣fi(k, kt)∣∣ e2π(λ′t+µ)|k|

≤ C(λ, λ′, κ)

(µ− µ′)dsupt≥0

(∑

k∈Zd

∣∣fi(k, kt)∣∣ e2π(λt+µ)|k|

).

Equivalently,

(4.45) supt≥0

∥∥ρ(t, · )∥∥Fλ′t+µ′ ≤ C sup

t≥0

∥∥∥∥∫fi S0

−t dv

∥∥∥∥Fλt+µ

.

By Propositions 4.15 and 4.19,∥∥∥∥∫fi S0

−t dv

∥∥∥∥Fλt+µ

≤∥∥fi S0

−t

∥∥Zλ,µ;1

t= ‖fi‖Zλ,µ;1

0≤ δ.

This and (4.45) imply (4.43).


To deduce (4.42), we first write

f(t, · ) = fi S0−t +

∫ t

0

((∇W ∗ ρτ ) S0

−(t−τ)

)· ∇vf

0 dτ,

where ρτ = ρ(τ, · ). Then for any λ′′ < λ′ we have, by Propositions 4.24 and 4.15,for all t ≥ 0,

‖f‖Zλ′′,µ′;1

t≤∥∥fi S0

−t

∥∥Zλ′′,µ;1

t+

∫ t

0

∥∥∥(∇W ∗ ρτ ) S0−(t−τ)

∥∥∥Zλ′′,µ′;∞

t

‖∇vf0‖

Zλ′′,µ;1t

dτ

(4.46)

= ‖fi‖Zλ′′,µ;1 +

(∫ t

0

‖∇W ∗ ρτ‖Fλ′′τ+µ′ dτ

)‖∇vf

0‖Cλ′′;1 .

Since ∇W (0) = 0, we have, for any τ ≥ 0,∥∥∇W ∗ ρτ

∥∥Fλ′′τ+µ ≤ e−2π(λ′′−λ′)τ

∥∥∇W ∗ ρτ

∥∥Fλ′τ+µ′

≤ ‖∇W‖L1 e−2π(λ′′−λ′)τ ‖ρτ‖Fλ′τ+µ′

≤ CW C δ e−2π(λ′′−λ′)τ ;

in particular

(4.47)

∫ t

0

∥∥∇W ∗ ρτ

∥∥Fλ′′+µ′ ≤ C δ

λ′′ − λ′.

Also, by Proposition 4.10, for 1 < λ′/λ′′ ≤ 2 we have

(4.48) ‖∇vf0‖Cλ′′;1 ≤

C

λ− λ′′‖f 0‖Cλ;1 ≤ C C0

λ− λ′′.

Plugging (4.47) and (4.48) in (4.46), we deduce (4.42). The end of the proof is aneasy exercise if one recalls that 〈f(t, · )〉 = 〈fi〉 for all t.

5. Deflection estimates

Let be given a small time-dependent force field, denoted by ε F (t, x), on Td ×Rd,whose analytic regularity improves linearly in time. (Think of εF as the force createdby a damped density.) This force field perturbs the trajectories S0

τ,t of the freetransport (τ the initial time, t the current time) into trajectories Sτ,t. The goal of thissection is to get an estimate on the maps Ωt,τ = St,τ S0

τ,t (so that St,τ = Ωt,τ S0t,τ).

These bounds should be in an analytic class about as good as F , with a loss ofanalyticity depending on ε; they should also be (for 0 ≤ τ ≤ t)

• uniform in t ≥ τ ;


• small as τ → ∞;

• small as τ → t.

We shall call Ωt,τ the deflection map (from time τ to time t). The idea isto compare the free (= without interaction) evolution to the true evolution is atthe basis of the interaction representation, wave operators, and most famously thescattering transforms (in which the whole evolution from t → −∞ to t → +∞ isreplaced by well-chosen asymptotics). For all these methods which are of constantuse in classical and quantum physics one can consult [21].

Remark 5.1. The order of composition in Ωt,τ = St,τ S0τ,t is the only one which

leads to useful asymptotics: it is easy to check that in general S0τ,t St,τ does not

converge to anything as t→ ∞, even if the force is compactly supported in time.

5.1. Formal expansion. Before stating the main result, we sketch a heuristic per-turbation study. Let us write a formal expansion of V0,t(x, v) as a perturbationseries:

V0,t(x, v) = v + ε v(1)(t, x, v) + ε2 v(2)(t, x, v) + . . .

Then we deduce

X0,t(x, v) = x+ vt+ ε

∫ t

0

v(1)(s, x, v) ds+ ε2

∫ t

0

v(2)(s, x, v) ds+ . . . ,

with v(i)(t = 0) = 0.So

∂2X0,t

∂t2= ε

∂v(1)

∂t+ ε2 ∂v

(2)

∂t+ . . . .

On the other hand,

ε F (t, X0,t) = ε∑

k

F (t, k) e2iπk·xe2iπk·vte2iπk·[εR t0 v(1) ds+ε2

R t0 v(2) ds+...]

= ε∑

k

F (t, k) e2iπk·xe2iπk·vt[1 + 2iπεk ·

∫ t

0

v(1) ds+ 2iπε2k ·∫ t

0

v(2) ds

− (2π)2ε2

(k ·∫ t

0

v(1) ds

)2

+ . . .].

By successive identification,

∂v(1)

∂t=∑

k

F (t, k) e2iπk·xe2iπk·vt;


∂v(2)

∂t=∑

k

F (t, k) e2iπk·xe2iπk·vt 2iπk ·∫ t

0

v(1) ds;

∂v(3)

∂t=∑

k

F (t, k) e2iπk·xe2iπk·vt

[2iπk ·

∫ t

0

v(2) ds− (2π)2ε2

(k ·∫ t

0

v(1) ds

)2],

etc.In particular notice that

∣∣∣∂v(1)

∂t

∣∣∣ ≤∑

k |F (t, k)|, so

∫ ∞

0

∣∣∣∣∂v(1)

∂t

∣∣∣∣ dt ≤∫ ∞

0

∑

k

|F (t, k)| dt ≤∫ ∞

0

∑|F (t, k)| e2πµte−2πµt dt

≤ CF

∫ ∞

0

e−2πµt =CF

2πµ.

So, under our uniform analyticity assumptions we expect V0,t(x, v) to be a uniformlybounded analytic perturbation of v.

5.2. Main result. On Tdx we consider the dynamical system

d2X

dt2= ε F (t, X);

its phase space is Td×Rd. Although this system is reversible, we shall only considert ≥ 0. The parameter ε is here only to recall the perturbative nature of the estimate.

For any (x, v) ∈ Td × Rd and any two times τ, t ∈ R+, let Sτ,t be the transformmapping the state of the system at time τ , to the state of the system at time t. Inmore precise terms, Sτ,t is described by the equations

Sτ,t(x, v) =(Xτ,t(x, v), Vτ,t(x, v)

);

Xτ,τ (x, v) = x, Vτ,τ (x, v) = v;

(5.1)d

dtXτ,t(x, v) = Vτ,t(x, v),

d

dtVτ,t(x, v) = ε F (t, Xτ,t(x, v)).

From the definition we have the composition identity

(5.2) St2,t3 St1,t2 = St1,t3 ;

in particular St,τ is the inverse of Sτ,t.We also write S0

τ,t for the same transform in the case of the free dynamics (ε = 0);in this case there is an explicit expression:

(5.3) S0τ,t(x, v) = (x+ v(t− τ), v),


where x + v(t − τ) is evaluated modulo Zd. Finally, we define the deflection mapassociated with εF :

(5.4) Ωt,τ = St,τ S0τ,t.

(There is no simple semigroup property for the transforms Ωt,τ .)In this section we establish the following estimates:

Theorem 5.2 (Analytic estimates on deflection in hybrid norms). Let ε > 0 andlet F = F (t, x) on R+ × Td satisfy

(5.5) F (t, 0) = 0, supt≥0

(‖F (t, ·)‖Fλt+µ + ‖∇xF (t, ·)‖Fλt+µ

)≤ CF

for some parameters λ, µ > 0 and CF > 0. Let t ≥ τ ≥ 0, and let

Ωt,τ =(ΩXt,τ ,ΩVt,τ

)

be the deflection associated with ε F . Let 0 ≤ λ′ < λ, 0 ≤ µ′ < µ and τ ′ ≥ 0 be suchthat

(5.6) λ′ (τ ′ − τ) ≤ (µ− µ′)

2.

Let R1(τ, t) = CF e

−2π (λ−λ′) τ min (t− τ) ; (2π(λ− λ′))−1 ;

R2(τ, t) = CF e−2π (λ−λ′) τ min (t− τ)2/2 ; (2π(λ− λ′))−2 .

Assume that

(5.7) ∀ 0 ≤ τ ≤ t, ε R2(τ, t) ≤(µ− µ′)

4,

and

(5.8) ε CF ≤ 4π2 (λ− λ′)2

2.

Then

(5.9) ∀ 0 ≤ τ ≤ t, ‖ΩXt,τ − Id ‖Zλ′,µ′

τ ′≤ 2 εR2(τ, t)

and

(5.10) ∀ 0 ≤ τ ≤ t, ‖ΩVt,τ − Id ‖Zλ′,µ′

τ ′≤ εR1(τ, t).

Remark 5.3. The proof of Theorem 5.2 is easily adapted to include Sobolev cor-rections. It is important to note that the deflection map is smooth, uniformly intime, not just in gliding regularity (τ ′ = 0 is admissible in (5.6)).


Proof of Theorem 5.2. For a start, let us make the ansatz

St,τ (x, v) =(x− v(t− τ) + ε Zt,τ (x, v), v + ε ∂τZt,τ (x, v)

),

with

Zt,t(x, v) = 0, ∂τZt,τ

∣∣∣τ=t

(x, v) = 0.

Then it is easily checked that

Ωt,τ − Id = ε (Z, ∂τZ) S0t−τ ;

in particular‖Ωt,τ − Id ‖

Zλ′,µ′

τ ′= ε

∥∥(Z, ∂τZ)∥∥Zλ′,µ′

t+τ ′−τ

.

To estimate this we shall use a fixed point argument based on the equation forSt,τ , namely

d2Xt,τ

dτ 2= ε F (τ,Xt,τ),

or equivalentlyd2Zt,τ

dτ 2= F

(τ, x− v(t− τ) + ε Zt,τ

).

So let us fix t and define

Ψ : (Wt,τ )0≤τ≤t 7−→ (Zt,τ )0≤τ≤t

such that (Zt,τ )0≤τ≤t is the solution of

(5.11)

∂2Zt,τ

∂τ 2= F

(τ, x− v(t− τ) + εWt,τ

)

Zt,t = 0, (∂τZt,τ )∣∣∣τ=t

= 0.

What we are after is an estimate of the fixed point of Ψ. We do this in two steps.

Step 1. Estimate of Ψ(0). Let Z0 = Ψ(0). By integration of (5.11) (for W = 0)we have

Z0t,τ =

∫ t

τ

(s− τ)F (s, x− v(t− s)) ds.

Let σ such that λ′σ ≤ (µ − µ′)/2. We apply the Zλ′,µ′

t+σ norm and use Proposi-tion 4.19:

‖Z0t,τ‖Zλ′,µ′

t+σ≤∫ t

τ

(s− τ) ‖F (s, · ) ‖Zλ′,µ′

s+σds =

∫ t

τ

(s− τ) ‖F (s, · ) ‖Fλ′s+λ′σ+µ′ ds.


Of course λ′σ + µ′ ≤ µ, so in particular

λ′ s+ λ′ σ + µ′ ≤ −(λ− λ′) s+ λ s+ µ.

Combining this with the assumption F (s, 0) = 0 yields

‖F (s, ·) ‖Fλ′s+λ′ σ+µ′ ≤ ‖F (s, ·) ‖Fλs+µ e−2π(λ−λ′) s

≤ CF e−2π(λ−λ′)s.

So

‖Z0t,τ‖Zλ′,µ′

t+σ≤ CF

∫ t

τ

(s− τ) e−2π(λ−λ′) s ds

≤ CF e−2π(λ−λ′) τ min

(t− τ)2

2;

1

(2π(λ− λ′))2

≤ R2(τ, t).

With t still fixed, we define the norm(5.12)∥∥∥∥∥∥∥∥ (Zt,τ )0≤τ≤t

∥∥∥∥∥∥∥∥ := sup

‖Zt,τ‖Zλ′,µ′

t+σ

R2(τ, t); 0 ≤ τ ≤ t; σ + t ≥ 0; λ′σ ≤ µ− µ′

2

.

The above estimates show that ‖‖Ψ(0)‖‖ ≤ 1. (We can assume t + σ ≥ 0 sincet+ (τ ′ − τ) ≥ t− τ ≥ 0, and we aim at finally choosing σ = τ ′ − τ .)

Step 2. Lipschitz constant of Ψ. We shall prove that under our assumptions,

Ψ is 1/2-Lipschitz on the ball B(0, 2) in the norm ‖‖ · ‖‖. Let W, W ∈ B(0, 2), and

Z = Ψ(W ), Z = Ψ(W ). By solving the differential inequality for Z − Z we get

Zt,τ − Zt,τ = ε

[∫ 1

0

∫ t

τ

(s− τ)∇xF(s, x− v(t− s)

+ ε(θWt,s + (1 − θ)Wt,s

))ds dθ

]·(Wt,s − Wt,s

).

We divide by R2(τ, t), take the Z norm, and note that R2(s, t) ≤ R2(τ, t); we get∥∥∥∥∥∥∥∥(Zt,τ − Zt,τ

)0≤τ≤t

∥∥∥∥∥∥∥∥ ≤ ε

∥∥∥∥∥∥∥∥(Wt,s − Wt,s

)0≤s≤t

∥∥∥∥∥∥∥∥A(t)


with

A(t) = supσ,τ

∫ 1

0

∫ t

τ

(s−τ)∥∥∥∥∥∇xF

(s, x−v(t−s)+ε

(θWt,s+(1−θ)Wt,s

))∥∥∥∥∥Zλ′,µ′

t+σ

ds dθ.

By Proposition 4.25 (composition inequality),

A(t) ≤∫ t

τ

(s− τ) ‖∇xF (s, · )‖Z

λ′,µ′+e(t,s,σ)s+σ

ds

=

∫ t

τ

(s− τ) ‖∇xF (s, · )‖Fλ′s+λ′σ+µ′+e(t,s,σ) ds,

with

e(t, s, σ) := ε∥∥∥θWt,s + (1 − θ) Wt,s

∥∥∥Zλ′,µ′

t+σ

≤ 2 εR2(s, t) ≤ 2 εR2(τ, t).

Using (5.7), we get

λ′s+ λ′σ + µ′ + e(s, t, σ) ≤ λ′s+ λ′σ + µ′ + 2 εR2(τ, t)

≤ λ′s+ µ = (λs+ µ) − (λ− λ′)s.

Using again the bound on ∇xF and the assumption F (s, 0) = 0, we deduce

A(t) ≤ supτ

∫ t

τ

(s− τ)CF e−2π(λ−λ′) s ds ≤ R2(0, t) ≤

CF

4π2 (λ− λ′)2.

Using (5.8), we conclude that∥∥∥∥∥∥∥∥(Zt,τ − Zt,τ

)0≤τ≤t

∥∥∥∥∥∥∥∥ ≤ 1

2

∥∥∥∥∥∥∥∥(Wt,s − Wt,s

)0≤s≤t

∥∥∥∥∥∥∥∥.

So Ψ is 1/2-Lipschitz on B(0, 2), and we can conclude the proof of (5.9) by applyingTheorem A.2 and choosing σ = τ ′ − τ .

It remains to control the velocity component of Ω, i.e., establish (5.10); thiswill follow from the control of the position component. Indeed, if we write Qt,τ =ε−1(ΩVt,τ − Id )(x, v), we have

Qt,τ =

∫ t

τ

F(s, x− v(t− s) + εWt,s

)ds

so we can estimate as before

‖Qt,τ‖Zλ′,µ′

t+(τ ′−τ)

≤∫ t

τ

‖F (s, ·)‖Fλ′s+λ′(τ ′−τ)+µ′+e(t,s,τ ′−τ) ds


to get

‖Qt,τ‖Zλ′,µ′

t+(τ ′−τ)

≤∫ t

τ

CF e−2π(λ−λ′) s ds ≤ R1(τ, t).

Thus the proof is complete.

Remark 5.4. Instead of directly studying St,τ S0τ,t one can study S0

t,τ Sτ,t andthen invert with the help of Proposition 4.28 (inversion inequality); in the end theresults are similar.

Remark 5.5. Loss and Bernard independently suggested to compare the estimatesin the present section with the Nekhoroshev theorem in dynamical systems theory[69, 70]. The latter theorem roughly states that for a perturbation of a completelyintegrable system, trajectories remain close to those of the unperturbed system fora time growing exponentially in the inverse of the size of the perturbation (unlikeKAM theory, this result is not global in time; but it is more general in the sense thatit also applies outside invariant tori). In the present setting the situation is bettersince the perturbation decays.

6. Bilinear regularity and decay estimates

To introduce this crucial section, let us reproduce and improve a key computationfrom Section 3. Let G be a function of v, and R a time-dependent function of xwith R(0) = 0; both G and R are vector-valued in Rd. (Think of G(v) as ∇vf(v)and of R(τ, x) as ∇W ∗ ρ(τ, x).) From now on we shall always use the supremumnorm over the coordinates in the following sense

‖F‖Z := max1≤i≤d

‖Fi‖Z , ‖F‖F := max1≤i≤d

‖Fi‖F .

Let further

σ(t, x) =

∫ t

0

∫

Rd

G(v) · R(τ, x− v(t− τ)

)dv dτ.

Then

σ(t, k) =

∫ t

0

∫

Td

∫

Rd

G(v) ·R(τ, x− v(t− τ)

)e−2iπk·x dv dx dτ

=

∫ t

0

∫

Td

∫

Rd

G(v) · R(τ, x) e−2iπk·x e−2iπk·v(t−τ) dv dx dτ

=

∫ t

0

G(k(t− τ)) · R(τ, k) dτ.


Let us assume that G has a “high” gliding analytic regularity λ, and estimate σin regularity λt, with λ < λ. Let α = α(t, τ) satisfy

0 ≤ α(t, τ) ≤ (λ− λ) (t− τ);

then

‖σ(t)‖Fλt ≤ d∑

k 6=0

∫ t

0

e2πλt|k| |G(k(t− τ))| |R(τ, k)| dτ

≤ d

∫ t

0

(supk 6=0

e2π[λ(t−τ)+α] |k| |G(k(t− τ))|) (∑

k

e2π(λτ−α)|k||R(τ, k)|)dτ

≤ d

(sup

ηe2πλ|η||G(η)|

) (sup

0≤τ≤t‖R(τ, ·)‖Fλτ−α

)∫ t

0

e−2π[(λ−λ)(t−τ)−α] dτ,

where we have used

k 6= 0 =⇒ 2π(λ(t− τ) + α)|k| ≤ 2πλ|k|(t− τ) −((λ− λ)(t− τ) − α

).

Let us choose

α(t, τ) =(λ− λ)

2min1 ; t− τ;

then ∫ t

0

e−2π[(λ−λ)(t−τ)−α

]dτ ≤

∫ t

0

e−π(λ−λ)(t−τ) dτ ≤ 1

π(λ− λ).

So in the end

‖σ(t)‖Fλt ≤ d ‖G‖Xλ

π(λ− λ)sup

0≤τ≤t‖R(τ)‖Fλτ−α(t,τ),

where ‖G‖Xλ = supη(e2πλ|η||G(η)|).

In the preceding computation there are three important things to notice, whichlie at the heart of Landau damping:

• The natural index of analytic regularity of σ in x increases linearly in time:this is an automatic consequence of the gliding regularity, already observedin Section 4.

• A bit α(t, τ) of analytic regularity of G was transferred from G to R, howevernot more than a fraction of (λ − λ)(t − τ). We call this the regularityextortion: if f forces f , it satisfies an equation of the form ∂tf + v · ∇xf +F [f ] · ∇vf = S, then f will give away some (gliding) smoothness to ρ =∫f dv.


• The combination of higher regularity of G and the assumption R(0) = 0has been converted into a time decay, so that the time-integral is bounded,uniformly as t→ ∞. Thus there is decay by regularity.

The main goal of this section is to establish quantitative variants of these effectsin some general situations when G is not only a function of v and R not only afunction of t, x. Note that we shall have to work with regularity indices dependingon t and τ !

Regularity extortion is related to velocity averaging regularity, well-known in ki-netic theory [42]; what is unusual though is that we are working in analytic regularity,and in large time, while velocity averaging regularity is mainly a short-time effect.In fact we shall study two distinct mechanisms for the extortion: the first one willbe well suited for short times (t − τ small), and will be crucial later to get rid ofsmall deteriorations in the functional spaces due to composition; the second one willbe well adapted to large times (t− τ → ∞) and will ensure convergence of the timeintegrals.

The estimates in this section lead to a serious twist on the popular view on Landaudamping, according to which the waves gives energy to the particles that it forces;instead, the picture here is that the wave gains regularity from the background, andregularity is converted into decay.

For the sake of pedagogy, we shall first establish the basic, simple bilinear estimate,and then discuss the two mechanisms once at a time.

6.1. Basic bilinear estimate.

Proposition 6.1 (Basic bilinear estimate in gliding regularity). Let G = G(τ, x, v),R = R(τ, x, v) be valued in Rd,

β(τ, x) =

∫

Rd

(G · R)(τ, x− v(t− τ), v

)dv,

σ(t, x) =

∫ t

0

β(τ, x) dτ.

Then

(6.1) ‖β(τ, ·)‖Fλt+µ ≤ d‖G‖Zλ,µ;1τ

‖R‖Zλ,µτ

;

and

(6.2) ‖σ(t, ·)‖Fλt+µ ≤ d

∫ t

0

‖G‖Zλ,µ;1τ

‖R‖Zλ,µτdτ.


Proof of Proposition 6.1. Obviously (6.2) follows from (6.1). To prove (6.1) we applysuccessively Propositions 4.15, 4.19 and 4.24:

‖β(τ, ·)‖Fλt+µ ≤∥∥∥∥∫

Rd

(G ·R) S0τ−t dv

∥∥∥∥Fλt+µ

≤∥∥∥(G · R) S0

τ−t

∥∥∥Zλ,µ;1

t

= ‖G · R‖Zλ,µ;1τ

≤ d ‖G‖Zλ,µ;1τ

‖R‖Zλ,µτ.

6.2. Short-term regularity extortion by time cheating.

Proposition 6.2 (Short-term regularity extortion). Let G = G(x, v), R = R(x, v)be valued in Rd, and

β(x) =

∫

Rd

(G · R) (x− v(t− τ), v) dv.

Then for any λ, µ, t ≥ 0 and any b > −1, we have

(6.3) ‖β‖Fλt+µ ≤ d ‖G‖Z

λ(1+b),µ;1

τ− bt1+b

‖R‖Z

λ(1+b),µ

τ− bt1+b

.

Moreover, if Pk stands for the projection on the kth Fourier mode as in (4.36), onehas

(6.4) e2π(λt+µ)|k||β(k)| ≤ d∑

ℓ∈Zd

‖PℓG‖Zλ(1+b),µ;1

τ− bt1+b

‖Pk−ℓR‖Zλ(1+b),µ

τ− bt1+b

.

Remark 6.3. If R only depends on t, x, then the norm of R in the right-hand sideof (6.3) is ‖R‖Fν with

ν = λ(1 + b)

∣∣∣∣τ −bt

1 + b

∣∣∣∣+ µ = (λτ + µ) − b(t− τ),

as soon as τ ≥ bt/(1 + b). Thus some regularity has been gained with respect toProposition 6.1. Even if R is not a function of t, x alone, but rather a function of t, xcomposed with a function depending on all the variables, this gain will be preservedthrough the composition inequality.


Proof of Proposition 6.2. By applying successively Propositions 4.15 (iv), 4.19 and4.24 (i),

∥∥∥∥∫

(G · R)(x− v(t− τ), v

)dv

∥∥∥∥Fλt+µ

≤∥∥∥(G · R)

(x− v(t− τ, v)

)∥∥∥Z

λ(1+b),µ;1t/(1+b)

=∥∥G · R‖

Zλ(1+b),µ;1

τ− bt1+b

≤ d ‖G‖Z

λ(1+b),µ;1

τ− bt1+b

‖R‖Z

λ(1+b),µ;∞

τ− bt1+b

,

which is the desired result (6.3).Inequality (6.4) is obtained in a similar way with the help of Proposition 4.34.

Remark 6.4. Let us sketch an alternative proof of Proposition 6.2, which is longerbut has the interest to rely on commutators involving ∇v, ∇x and the transportsemigroup, all of them classically related to hypoelliptic regularity and velocity av-eraging. Let S = S0

τ−t, so that R S(x, v) = R(x− v(t− τ), v); and let

D = Dτ,t,b :=(τ − b(t− τ)

)∇x + (1 + b)∇v.

Then by direct computation

(6.5) t∇x(R S) = (DR) S − (1 + b)∇v(R S);

and since ∇x commutes with ∇v and D, and with the composition by S as well, thiscan be generalized by induction into

(6.6) (t∇x)n(R S) =∑

m≤n

( nm

)[−(1 + b)∇v

]m((Dn−mR) S

).

Applying this formula with R replaced by G · R and integrating in v yields

(t∇x)n

∫

Rd

(G · R) S0τ−t dv =

∫

Rd

Dn(G ·R) S0τ−t dv

=

∫

Rd

Dn(G ·R) dv.

It follows by taking Fourier transform that

(2iπtk)nβ(k) =

∫

Rd

[Dn(G · R)

]bdv

=

∫

Rd

((1 + b)∇v + 2iπ

(τ − b(t− τ)

)k)n

(G ·R)b(k, v) dv,


whence∑

k,n

e2πµ|k| |2πλtk|nn!

|β(k)|

≤∑

k,n

e2πµ|k|

(λ(1 + b)

)n

n!

∥∥∥[∇v + 2iπ

(τ − bt

1 + b

)k]n

(G · R)b(k, v)∥∥∥

L1(dv)

= ‖G · R‖Z

λ(1+b),µ;1

τ− bt1+b

,

and then (6.3) follows by Proposition 4.24.

Let us conclude this subsection with some comments on Proposition 6.2. Whenwe wish to apply it, what constraints on b(t, τ) (assumed to be nonnegative to fixthe ideas) does this presuppose? First, b should be small, so that λ(1+b) ≤ λ given.But most importantly, we have estimated Gτ in a norm Zτ ′ instead of Zτ (this isthe time cheating), where |τ ′ − τ | = bt/(1 + b). To compensate for this discrepancy,we may apply (4.19), but for this to work bt/(1 + b) should be small, otherwise wewould lose a large index of analyticity in x, or at best we would inherit an undesirableexponentially growing constant. So all we are allowed is b(t, τ) = O(1/(1+ t)). Thisis not enough to get the time-decay which would lead to Landau damping. Indeed,

if R = R(x) with R(0) = 0, then

‖R‖Z

λ(1+b),µτ−bt/(1+b)

= ‖R‖Fλτ+µ−λ b(t−τ) ≤ e−λb(t−τ) ‖R‖Fλτ+µ;

so we gain a coefficient e−λb(t−τ), but then∫ t

0

e−λb(t−τ) dτ ≥∫ t

0

e−λε( t−τt ) dτ =

(1 − e−λε

λ ε

)t,

which of course diverges in large time.To summarize: Proposition 6.2 is helpful when (t−τ) = O(1), or when some extra

time-decay is available. This will already be very useful; but for long-time estimateswe need another, complementary mechanism.

6.3. Long-term regularity extortion. To search for the extra decay, let us refinethe computation of the beginning of this section. Assume that Gτ = ∇vgτ , where(gτ)τ≥0 solves a transport-like equation, so G(τ, k, η) = 2iπη g(τ, k, η), and

|G(τ, k, η)| . 2π|η| e−2πµ|k| e−2πλ|η+kτ |.


Up to slightly increasing λ and µ, we may assume

(6.7) |G(τ, k, η)| . (1 + τ) e−2πµ|k| e−2πλ|η+kτ |.

Let then ρ(τ, x) =∫f(τ, x, v) dv, where also f solves a transport equation, but has

a lower analytic regularity; and R = ∇W ∗ ρ. Assuming |∇W (k)| = O(|k|−γ) forsome γ ≥ 0, we have

(6.8) |R(τ, k)| . e−2π(λτ+µ)|k| 1k 6=0

1 + |k|γ .

Let again

σ(t, x) =

∫ t

0

∫

Rd

G(τ, x− v(t− τ), v

)· R(τ, x− v(t− τ)

)dv dτ.

As t → +∞, G in the integrand of σ oscillates wildly in phase space, so it is notclear that it will help at all. But let us compute:

σ(t, k) =

∫ t

0

∫

Rd

∫

Td

G(τ, x− v(t− τ), v

)· R(τ, x− v(t− τ)

)e−2iπk·x dx dv dτ

=

∫ t

0

∫

Rd

∫

Td

G(τ, x, v) ·R(τ, x) e−2iπk·x e−2iπk·v(t−τ) dx dv dτ

=

∫ t

0

∫

Rd

G ·R(τ, k, v) e−2iπk·v(t−τ) dv dτ

=

∫ t

0

∫

Rd

∑

ℓ

G(τ, ℓ, v) · R(τ, k − ℓ) e−2iπk·v(t−τ) dv dτ

=

∫ t

0

∑

ℓ

G(τ, ℓ, k(t− τ)

)· R(τ, k − ℓ) dτ.

At this level, the difference with respect to the beginning of this section lies in thefact that there is a summation over ℓ ∈ Zd, instead of just choosing ℓ = 0. Notethat σ(t, 0) =

∫ t

0

∫∫G(τ, x, v) · R(τ, x) dx dv dτ = 0, because G is a v-gradient.

From (6.7) and (6.8) we deduce∑

k

e2π(λt+µ)|k||σ(t, k)|

.

∫ t

0

(1 + τ)∑

ℓ 6=k, k 6=0

e2πµ|k| e2πλt|k| e−2πµ|ℓ| e−2πλ|k(t−τ)+ℓτ | e−2πµ|k−ℓ| e−2πλτ |k−ℓ|

1 + |k − ℓ|γ .


Using the inequalities

e−2πµ|k−ℓ| e2πµ|k| e−2πµ|ℓ| ≤ e−2π(µ−µ)|ℓ|

ande−2πλτ |k−ℓ| e2πλt|k| e−2πλ|k(t−τ)+ℓτ | ≤ e−2π(λ−λ)|k(t−τ)+ℓτ |,

we end up with

‖σ(t)‖Fλt+µ .∑

k 6=0, ℓ 6=k

e−2π(µ−µ)|ℓ|

1 + |k − ℓ|γ∫ t

0

e−2π(λ−λ)|k(t−τ)+ℓτ | (1 + τ) dτ.

If it were not for the negative exponential, the time-integral would be O(t2) ast → ∞. The exponential helps only a bit: its argument vanishes e.g. for d = 1,k > 0, ℓ < 0 and τ = (k/(k + |ℓ|))t. Thus we have the essentially optimal bounds

(6.9)

∫ t

0

e−2π(λ−λ)|k(t−τ)+ℓτ | dτ ≤ 1

π(λ− λ) |k − ℓ|and

(6.10)

∫ t

0

e−2π(λ−λ)|k(t−τ)+ℓτ | τ dτ ≤ 1

2π2(λ− λ)2 |k − ℓ|2+

(1

π(λ− λ)

) |k|t|k − ℓ| .

From this computation we conclude that:

• The higher regularity of G has allowed to reduce the time-integral thanks to afactor e−α|k(t−τ)+ℓτ |; but this factor is not small when τ/t is equal to k/(k − ℓ). Asdiscussed in the next section, this reflects an important physical phenomenon called(plasma) echo, which can be assimilated to a resonance.

• If we had (in “gliding” norm) ‖Gτ‖ = O(1) this would ensure a uniform boundon the integral, as soon as γ > 0, thanks to (6.9) and

∑

k,ℓ

e−α|ℓ|

(1 + |k − ℓ|)1+γ< +∞.

• But Gτ is a velocity-gradient, so — unless of course G depends only on v —‖Gτ‖ diverges like O(τ) as τ → ∞, which implies a divergence of our bounds in largetime, as can be seen from (6.10). If γ ≤ 1 this comes with a divergence in the k

variable, since in this case∑

k,ℓe−α|ℓ||k|

(1+|k−ℓ|)1+γ = +∞. (The Coulomb case corresponds

to γ = 1, so in this respect it has a borderline divergence.)

The following estimate adapts this computation to the formalism of hybrid norms,and at the same time allows a time-cheating similar to the one in Proposition 6.2.


Fortunately, we shall only need to treat the case when R = R(τ, x); the more generalcase with R = R(τ, x, v) would be much more tricky.

Theorem 6.5 (Long-term regularity extortion). Let G = G(τ, x, v), R = R(τ, x),and

σ(t, x) =

∫ t

0

∫

Rd

G(τ, x− v(t− τ), v

)· R(τ, x− v(t− τ)

)dv dτ.

Let λ, λ, µ, µ, µ′ = µ′(t, τ), M ≥ 1 such that (1+M)λ ≥ λ > λ > 0, µ ≥ µ′ > µ > 0,γ ≥ 0 and b = b(t, τ) ≥ 0. Then

(6.11) ‖σ(t, ·)‖Fλt+µ ≤∫ t

0

KG0 (t, τ) ‖Rτ‖Fν dτ +

∫ t

0

KG1 (t, τ) ‖Rτ‖Fν,γ dτ,

where

(6.12) ν = max

λτ + µ′ − λ

2b(t− τ) ; 0

,

(6.13) KG0 (t, τ) = d e

−2π“

λ−λ2

”(t−τ)

∥∥∥∥∫G(τ, x, · ) dx

∥∥∥∥Cλ(1+b);1

,

(6.14) KG1 (t, τ) = sup

0≤τ≤t

‖Gτ‖Zλ(1+b),µ

τ−bt/(1+b)

1 + τ

K1(t, τ),

(6.15)

K1(t, τ) = (1+τ) d supk 6=0, ℓ 6=0

e

−2π(µ−µ2 )|ℓ| e

−2π“

λ−λ2M

”|k(t−τ)+ℓτ |

e−2π[(µ′−µ)+ λ b

2(t−τ)

]|k−ℓ|

1 + |k − ℓ|γ

.

Remark 6.6. It is essential in (6.11) to separate the contribution of G(τ, 0, v) fromthe rest. Indeed, if we removed the restriction ℓ 6= 0 in (6.15) the kernel K1 wouldbe too large to be correctly controlled in large time. What makes this separationreasonable is that, although in cases of application G(τ, x, v) is expected to grow likeO(τ) in large time, the spatial average

∫G(τ, x, v) dx is expected to be bounded.

Also, we will not need to take advantage of the parameter γ to handle this term.

Proof of Theorem 6.5. Without loss of generality we may assume that G and R arescalar-valued. (This explains the constant d in front of the right-hand sides of (6.13)


and (6.15).) First we assume G(τ, 0, v) = 0, and we write as before

σ(t, k) =

∫ t

0

∑

ℓ∈Zd\0

∫

Rd

G(τ, ℓ, v) R(τ, k − ℓ) e−2iπk·v(t−τ) dv

dτ,

(6.16) |σ(t, k)| ≤∫ t

0

∑

ℓ 6=0

∣∣∣∣∫G(τ, ℓ, v) e−2iπk·v(t−τ) dv

∣∣∣∣ |R(τ, k − ℓ)| dτ.

Next we let τ ′ = τ − b(t− τ) and write

(6.17) e2π(λt+µ)|k| ≤ e−2π(µ−µ)|ℓ| e−2πλ(τ−τ ′)|k−ℓ| e−2π(µ′−µ)|k−ℓ| e−2π(λ−λ) |k(t−τ ′)+ℓτ ′|

e2πµ|ℓ| e2π(λτ+µ′)|k−ℓ| e2πλ|k(t−τ ′)+ℓτ ′|.

Since 0 ≤ λ− λ ≤Mλ, we have

e−2π(λ−λ)|k(t−τ ′)+ℓτ ′| ≤ e−2π

“λ−λ2M

”|k(t−τ)+ℓτ |

e2π λ2(τ−τ ′)|k−ℓ|;

so (6.17) implies

(6.18) e2π(λt+µ)|k| ≤ e−2π(µ−µ)|ℓ| e−2π

“λ−λ2M

”|k(t−τ)+ℓτ |

e−2π[(µ′−µ)+ λ

2(τ−τ ′)

]|k−ℓ|

e2πµ|ℓ| e2π(λτ+µ′)|k−ℓ|∑

n∈Nd0

∣∣(2iπλ)(k(t− τ ′) + ℓτ ′

)∣∣n

n!.


For each n ∈ Nd0,∣∣∣(2iπλ)

(k(t− τ ′) + ℓτ ′

)∣∣∣n

n!

∣∣∣∣∫G(τ, ℓ, v) e−2iπk·v(t−τ) dv

∣∣∣∣

=λ

n

n!

∣∣∣∣∫G(τ, ℓ, v)

[2iπ(k(t− τ ′) + ℓτ ′)

]ne−2iπk·v(t−τ) dv

∣∣∣∣

=λ

n

n!

(t− τ ′

t− τ

)n ∣∣∣∣∫G(τ, ℓ, v)

[(2iπ)

(k(t− τ) + ℓτ ′

(t− τ

t− τ ′

))]n

e−2iπk·v(t−τ) dv

∣∣∣∣

=λ

n

n!

(t− τ ′

t− τ

)n ∣∣∣∣∫G(τ, ℓ, v)

[−∇v + 2iπℓτ ′

(t− τ

t− τ ′

)]n

e−2iπk·v(t−τ) dv

∣∣∣∣

=λ

n

n!

(t− τ ′

t− τ

)n ∣∣∣∣∫ [

∇v + 2iπℓτ ′(t− τ

t− τ ′

)]n

G(τ, ℓ, v) e−2iπk·v(t−τ) dv

∣∣∣∣

≤ λn

n!

(t− τ ′

t− τ

)n ∥∥∥∥(∇v + 2iπℓτ ′

(t− τ

t− τ ′

))n

G(τ, ℓ, v)

∥∥∥∥L1(dv)

=λ

n(1 + b)n

n!

∥∥∥∥(∇v + 2iπℓ

(τ − bt

1 + b

))n

G(τ, ℓ, v)

∥∥∥∥L1(dv)

.

Combining this with (6.16) and (6.18), summing over k, we deduce∥∥σ(t, · )

∥∥Fλt+µ =

∑

k 6=0

e2π(λt+µ)|k| |σ(t, k)|

≤∫ t

0

∑

kℓn

e

−2π(µ−µ)|ℓ| e−2π

“λ−λ2M

”|k(t−τ)+ℓτ |

e−2π((µ′−µ)+ λ

2(τ−τ ′)

)|k−ℓ|

1 + |k − ℓ|γ

e2πµ|ℓ| e2π

[(λτ+µ′)−λ

2b(t−τ)

]|k−ℓ|

λn(1 + b)n

n!|R(τ, k − ℓ)|

∥∥∥∥(∇v + 2iπℓ

(τ − bt

1 + b

))n

G(τ, ℓ, v)

∥∥∥∥L1(dv)

dτ,

and the desired estimate follows readily.

Finally we consider the contribution of G(τ, 0, v) =∫G(τ, x, v) dx. This is done

in the same way, noting that

supk 6=0

e−2π(λ−λ)|k|(t−τ)

1 + |k|γ ≤ e−2π(λ−λ)(t−τ).


To conclude this section we provide a “mode by mode” variant of Theorem 6.5;this will be useful for very singular interactions (γ = 1 in Theorem 2.6).

Theorem 6.7. Under the same assumptions as Theorem 6.5, for all k ∈ Zd we havethe estimate

(6.19) e2π(λt+µ)|k||σ(t, k)| ≤∫ t

0

KG0 (t, τ)

(e2πν|k| |R(τ, k)|

)dτ

+

∫ t

0

∑

ℓ∈Zd

KGk,ℓ(t, τ) e

2πν|k−ℓ| (1 + |k − ℓ|γ) |R(τ, k − ℓ)| dτ,

where KG0 is defined by (6.13), ν by (6.12), and

KGk,ℓ(t, τ) = sup

0≤τ≤t

‖G‖

Zλ(1+b),µ;1τ−bt/(1+b)

1 + τ

Kk,ℓ(t, τ),

Kk,ℓ(t, τ) =(1 + τ) d e−2π(µ−µ)|ℓ| e

−2π“

λ−λ2M

”|k(t−τ)+ℓτ |

e−2π[(µ′−µ)+ λ

2b(t−τ)

]|k−ℓ|

1 + |k − ℓ|γ .

Proof of Theorem 6.7. The proof is similar to the proof of Theorem 6.5, except thatk is fixed and we use, for each ℓ, the crude bound

e2πµ|ℓ|

∥∥∥∥[∇v + 2iπℓ

(τ − bt

1 + b

)]n

G(τ, ℓ, v)

∥∥∥∥L1(dv)

≤∑

j∈Zd

e2πµ|j|

∥∥∥∥[∇v + 2iπj

(τ − bt

1 + b

)]n

G(τ, j, v)

∥∥∥∥L1(dv)

.

7. Control of the time-response

To motivate this section, let us start from the linearized equation (3.3), but nowassume that f 0 depends on t, x, v and that there is an extra source term S, decayingin time. Thus the equation is

∂f

∂t+ v · ∇xf − (∇W ∗ ρ) · ∇vf

0 = S,


and the equation for the density ρ, as in the proof of Theorem 3.1, is

(7.1)

ρ(t, x) =

∫

Rd

fi(x−vt, v) dv+∫ t

0

∫

Rd

∇vf0(τ, x−v(t−τ), v

)·(∇W∗ρ)(τ, x−v(t−τ)) dv dτ

+

∫ t

0

∫

Rd

S(τ, x− v(t− τ), v) dv dτ.

Hopefully we may apply Theorem 6.5 to deduce from (7.1) an integral inequalityon ϕ(t) := ‖ρ(t)‖Fλt+µ, which will look like

(7.2) ϕ(t) ≤ A + c

∫ t

0

K(t, τ)ϕ(τ) dτ,

where A is the contribution of the initial datum and the source term, and K(t, τ) akernel looking like, say, (6.15).

From (7.2) how do we proceed? Assume for a start that a smallness condition ofthe form (a) in Proposition 2.1 is satisfied. Then the simple and natural way, as inSection 3, would be to write

ϕ(t) ≤ A + c

(∫ t

0

K(t, τ) dτ

) (sup

0≤τ≤tϕ(τ)

),

and deduce

(7.3) ϕ(t) ≤ A(1 − c

∫ t

0K(t, τ) dτ

)

(assuming of course the denominator to be positive). However, if K is given by

(6.15), it is easily seen that∫ t

0K(t, τ) dτ ≥ κ t as t → ∞, where κ > 0; then

(7.3) is useless. In fact (7.2) does not prevent ϕ from going to +∞ as t → ∞.Nevertheless, its growth may be controlled under certain assumptions, as we shallsee in this section. Before embarking on cumbersome calculations, we shall startwith a qualitative discussion.

7.1. Qualitative discussion. The kernel K in (6.15) depends on the choice ofµ′ = µ(t, τ). How large µ′−µ can be depends in turn on the amount of regularizationoffered by the convolution with the interaction ∇W . We shall distinguish severalcases according to the regularity of the interaction.


7.1.1. Analytic interaction. If ∇W is analytic, there is σ > 0 such that

∀ν ≥ 0, ‖ρ ∗ ∇W‖Fν+σ ≤ C ‖ρ‖Fν ;

then in (6.15) we can afford to choose, say, µ′ − µ = σ, and γ = 0. Thus, assumingb = B/(1+ t) with B small enough so that (µ′−µ)−λb(t− τ) ≥ σ/2, K is boundedby

(7.4) K(α)

(t, τ) = (1 + τ) supk 6=0, ℓ 6=0

e−α|ℓ| e−α|k−ℓ| e−α|k(t−τ)+ℓτ |,

where α = 12minλ − λ ; µ − µ ; σ. To fix ideas, let us work in dimension d = 1.

The goal is to estimate solutions of

(7.5) ϕ(t) ≤ a + c

∫ t

0

K(α)

(t, τ)ϕ(τ) dτ.

Whenever τ/t is a rational number distinct from 0 or 1, there are k, ℓ ∈ Z such

that |k(t− τ) + ℓτ | = 0, and the size of K(α)

(t, τ) mainly depends on the minimumadmissible values of k and k − ℓ. Looking at values of τ/t of the form 1/(n+ 1) orn/(n+ 1) suggests the approximation

(7.6) K(α)

. (1 + τ) mine−α( τ

t−τ ) e−2α ; e−2α( t−ττ ) e−α

.

But this estimate is terrible: the time-integral of the right-hand side is much larger

than the integral of K(α)

. In fact, the fast variation and “wiggling” behavior of K(α)

are essential to get decent estimates.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

exact t = 10approx t = 10

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


0

20

40

60

80

100

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Figure 5. the kernel K(α)

(t, τ), together with the approximate upperbound in (7.6), for α = 0.5 and t = 10, t = 100, t = 1000.


To get a better feeling for K(α)

, let us only retain the term in k = 1, ℓ = −1; thisseems reasonable since we have an exponential decay as k or ℓ go to infinity (anyway,throwing away all other terms can only improve the estimates). So we look at

K(α)(t, τ) = (1+ τ) e−3α e−α|t−2τ |. Let us time-rescale by setting kt(θ) = t K(α)(t, tθ)for θ ∈ [0, 1] (the t factor because dτ = t dθ); then it is not hard to see that

kt

t−→ e−3α

2αδ 1

2

This suggests the following baby model for (7.5):

(7.7) ϕ(t) ≤ a + c t ϕ

(t

2

).

The important point in (7.7) is that, although the kernel has total mass O(t), thismass is located far from the endpoint τ = t; this is what prevents the fast growthof ϕ. Compare with the inequality ϕ(t) ≤ a + c t ϕ(t), which implies no restrictionat all on ϕ.

To be slightly more quantitative, let us look for a power series Φ(t) =∑

k ak tk

achieving equality in (7.7). This yields a0 = a, ak+1 = c ak 2−k, so

(7.8) Φ(t) = a∞∑

k=0

ck tk

2k(k−1)/2.

The function Φ exhibits a truly remarkable behavior: it grows faster than any poly-nomial, but slower than any fractional exponential exp(c tν), ν ∈ (0, 1); essentially

it behaves like A(log t)2 (as can also be seen directly from (7.7)). One may conjecturethat solutions of (7.5) exhibit a similar kind of growth.

Let us interpret these calculations. Typically, the kernel K controls the timevariation of (say) the spatial density ρ which is due to binary interaction of waves.When two waves of distinct frequencies interact, the effect over a long time periodis most of the time very small; this is a consequence of the oscillatory nature ofthe evolution, and the resulting time-averaging. But at certain particular times, theinteraction becomes strong: this is known in plasma physics as the plasma echo,and can be thought of as a kind of resonance. Spectacular experiments by Malmbergand collaborators are based on this effect [32, 58]. Namely, if one starts a wave atfrequency ℓ at time 0, and forces it at time τ by a wave of frequency k− ℓ, a strongresponse is obtained at time t and frequency k such that

(7.9) k(t− τ) + ℓτ = 0


(which of course is possible only if k and ℓ are parallel to each other, with oppositedirections).

In the present nonlinear setting, whatever variation the density function is subjectto, will result in echoes at later times. Even if each echo in itself will eventuallydecay, the problem is whether the accumulation of echoes will trigger an uncontrolledgrowth (unstability). As long as the expected growth is eaten by the time-decaycoming from the linear theory, nonlinear Landau damping is expected. In the presentcase, the growth of (7.8) is very slow in regard of the exponential time-decay due tothe analytic regularity.

7.1.2. Sobolev interaction. If ∇W only has Sobolev regularity, we cannot afford in(6.15) to take µ′(t, τ) larger than µ + η(t − τ)/t (because the amount of regularitytransferred in the bilinear estimates is only O((t− τ)/t), recall the discussion at theend of Subsection 6.2). On the other hand, we have γ > 0 such that

∀ν ≥ 0, ‖∇W ∗ ρ‖Fν,γ ≤ C ‖ρ‖Fν ,

and then we can choose this γ in (6.15). So, assuming b = B/(1 + t) with B smallenough so that (µ′−µ)−λb(t− τ) ≥ η(t− τ)/(2t), K in (6.15) will be controlled by

(7.10) K(α),γ(t, τ) = (1 + τ)d supk 6=0, ℓ 6=0

e−α|ℓ| e−α( t−τt )|k−ℓ| e−α|k(t−τ)+ℓτ |

1 + |k − ℓ|γ ,

where α = 12minλ− λ ; µ− µ ; η. The equation we are considering now is

(7.11) ϕ(t) ≤ a+

∫ t

0

K(α),γ(t, τ)ϕ(τ) dτ.

For, say, τ ≤ t/2, we have K(α) ≤ K(α/2)

, and the discussion is similar to that in7.1.1. But when τ aproaches t, the term exp

(−α( t−τ

t) |k− ℓ|

)hardly helps. Keeping

only k > 0 and ℓ = −1 (because of the exponential decay in ℓ) leads to consider thekernel (for simplicity we set d = 1 without loss of generality)

K(α)(t, τ) = (1 + τ) supk 6=0

e−α|kt−(k+1)τ |

1 + (k + 1)γ.

Once again we perform a time-rescaling, setting kt(θ) = t K(α)(t, tθ), and let t →∞. In this limit each exponential exp(−α|kt − (k + 1)τ |) becomes localized in aneighborhood of size O(1/k t) around θ = k/(k + 1), and contributes a Dirac mass


at θ = k/(k + 1), with amplitude 2/(α(k + 1));

kt

t−−−→t→∞

2

α

∑

k

1

1 + (k + 1)γ

k

(k + 1)2δ1− 1

k+1.

This leads us to the following baby model for (7.11):

(7.12) ϕ(t) ≤ a+ c t∑

k≥1

1

k1+γϕ

((1 − 1

k

)t

).

If we search for∑ant

n achieving equality, this yields

a0 = a, an+1 = c

(∑

k≥1

1

k1+γ

(1 − 1

k

)n)an.

To estimate the behavior of the∑

k above, we compare it with∫ ∞

1

1

t1+γ

(1 − 1

t

)n

dt =

∫ t

0

uγ−1 (1 − u)n du = B(γ, n+ 1) (Beta function)

=Γ(γ) Γ(n+ 1)

Γ(n + γ + 1)= O

(1

nγ

).

All in all, we may expect ϕ in (7.11) to behave qualitatively like

Φ(t) = a∑

n≥0

cn tn

(n!)γ.

Notice that Φ is subexponential for γ > 1 (it grows essentially like the fractionalexponential exp(t1/γ)) and exponential for γ = 1. In particular, as soon as γ > 1 weexpect nonlinear Landau damping again.

7.1.3. Coulomb/Newton interaction (γ = 1). When γ = 1, as is the case for Coulombor Newton interaction, the previous analysis becomes borderline since we expect(7.12) to be compatible with an exponential growth, and the linear decay is also ex-ponential. To handle this more singular case, we shall work mode by mode, ratherthan on just one norm. Starting again from (7.1), we consider, for each k ∈ Zd,

ϕk(t) = e2π(λt+µ)|k| |ρ(t, k)|,and hope to get, via Theorem 6.7, an inequality which will roughly take the form

(7.13) ϕk(t) ≤ Ak + c

∫ t

0

∑

ℓ

Kk,ℓ(t, τ)ϕk−ℓ(τ) dτ.


(Note: summing in k would yield an inequality worse than (7.11).) To fix the ideas,let us work in dimension d = 1, and set k ≥ 1, ℓ = −1. Reasoning as in subsection(7.1.2), we obtain the baby model

(7.14) ϕk(t) ≤ Ak +c t

(k + 1)1+γϕk+1

(kt

k + 1

).

The gain with respect to (7.12) is clear: for different values of k, the “dominanttimes” are distinct. From the physical point of view, we are discovering that, in somesense, echoes occurring at distinct frequencies are asymptotically well separated.

Let us search again for power series solutions: we set

ϕk(t) =∑

m≥0

ak,m tm, ak,0 = Ak.

By identification, ak,m = ak+1,m−1 c (k + 1)−(1+γ) (k/(k + 1))m−1, and by induction

ak,m = Ak+m cm

[k!

(k +m)!

]1+γkm−1 cm

(k + 1)(k + 2) . . . (k +m)≃ Ak+m

[k!

(k +m)!

]γ+2

km−1 cm.

We may expect Ak+m . Ae−a(k+m); then

ak,m . A (k e−ak) km cme−am

(m!)γ+2,

and in particular

ϕk(t) . Ae−ak/2∑

m

(ckt)m

(m!)γ+2. Ae(1−α) (ckt)α

, α =1

γ + 2.

This behaves like a fractional exponential even for γ = 1, and we can now believe innonlinear Landau damping for such interactions! (The argument above works evenfor more singular interactions; but in the proof later the condition γ ≥ 1 will berequired for other reasons, see pp. 136 and 147.)

7.2. Exponential moments of the kernel. Now we start to estimate the kernelK(α),γ from (7.10), without any approximation this time. Eventually, instead ofproving that the growth is at most fractional exponential, we shall compare it witha slow exponential eεt. For this, the first step consists in estimating exponentialmoments of the kernel e−εt

∫K(t, τ) eετ dτ . (To get more precise estimates, one can

study e−εtα∫K(t, τ) eετα

dτ , but such a refinement is not needed for the proof ofTheorem 2.6.)

The first step consists in estimating exponential moments.


Proposition 7.1 (Exponential moments of the kernel). Let γ ∈ [1,∞) be given.For any α ∈ (0, 1), let K(α),γ be defined by (7.10). Then for any γ < ∞ there isα = α(γ) > 0 such that if α ≤ α and ε ∈ (0, 1), then for any t > 0,

e−εt

∫ t

0

K(α),γ(t, τ) eετ dτ

≤ C

(1

α εγ tγ−1+

ln 1α

α εγ tγ+

1

α2 ε1+γ t1+γ+

(1

α3+

ln 1α

α2ε

)e−

ε t4 +

e−α t2

α3

),

where C = C(γ). In particular,

• If γ > 1 and ε ≤ α, then e−εt

∫ t

0

K(α),γ(t, τ) eετ dτ ≤ C(γ)

α3 ε1+γ tγ−1;

• If γ = 1 then e−εt

∫ t

0

K(α),γ(t, τ) eετ dτ ≤ C

α3

(1

ε+

1

ε2 t

).

Remark 7.2. Much stronger estimates can be obtained if the interaction is analytic;

that is, when K(α),γ is replaced by K(α)

defined in (7.4). A notable point aboutProposition 7.1 is that for γ = 1 we do not have any time-decay as t→ ∞.

Proof of Proposition 7.1. To simplify notation we shall not recall the dependence ofK on γ, and we shall set d = 1 without loss of generality. We first assume γ < ∞,and consider τ ≤ t/2, which is the favorable case. We write

K(α)(t, τ) ≤ (1 + τ) supk 6=0

supℓe−α|ℓ| e−

α2|k−ℓ| e−α|k(t−τ)+ℓτ |.

Since we got rid of the condition ℓ 6= 0, the right-hand side is now a nonincreasingfunction of d. (To see this, pick up a nonzero component of k, and recall our normconventions from Appendix A.1.) So we assume d = 1. By symmetry we may alsoassume k > 0.


Explicit computations yield

∫ t/2

0

e−α|k(t−τ)+ℓτ | (1+τ) dτ ≤

1

α (ℓ− k)+

1

α2 (ℓ− k)2if ℓ > k

e−αkt

(t

2+t2

8

)if ℓ = k

e−α( k+ℓ2 )t

α|k − ℓ|

(1 +

t

2

)if −k ≤ ℓ < k

(2

α|k − ℓ| +2 kt

α|k − ℓ|2 +1

α2|k − ℓ|2)

if ℓ < −k.

In all cases,

∫ t/2

0

e−α|k(t−τ)+ℓτ | (1 + τ) dτ ≤(

3

α|k − ℓ| +1

α2|k − ℓ|2 +2 t

α|k − ℓ|

)1k 6=ℓ

+ e−αkt

(t

2+t2

8

)1ℓ=k.

So

e−εt

∫ t/2

0

e−α|k(t−τ)+ℓτ | (1 + τ) eετ dτ

≤ e−εt2

(3

α|k − ℓ| +1

α2|k − ℓ|2 +2 t

α|k − ℓ|

)1k 6=ℓ + e−αkt

(t

2+t2

8

)1ℓ=k

≤ e−εt4

(3

α|k − ℓ| +1

α2|k − ℓ|2 +8 z

αε|k − ℓ|

)1k 6=ℓ + e−

tα2

(z

α+

8 z2

α2

)1ℓ=k,

where z = sup(xe−x) = e−1. Then

eεt

∫ t/2

0

K(α)(t, τ) eετ dτ

≤ e−εt4

∑

k 6=0

∑

ℓ 6=k

e−α|ℓ| e−α2|k−ℓ|

(3

α|k − ℓ| +1

α2|k − ℓ|2 +8 z

αε|k − ℓ|

)

+ e−tα2

∑

ℓ

e−α|ℓ|

(z

α+

8z2

α2

).


Using the bounds (for α ∼ 0+)

∑

ℓ

e−αℓ = O

(1

α

),∑

ℓ

e−αℓ

ℓ= O

(ln

1

α

),∑

ℓ

e−αℓ

ℓ2= O(1),

we end up, for α ≤ 1/4, with a bound like

C e−εt4

(1

α2ln

1

α+

1

α3+

1

α2εln

1

α

)+ C e−

αt2

(1

α2+

1

α3

)

≤ C

[e−

εt4

(1

α3+

1

α2εln

1

α

)+e−

αt2

α3

].

(Note that the last term is O(t−3), so it is anyway negligible in front of the otherterms if γ ≤ 4; in this case the restriction ε ≤ α can be dispended with.)

• Next we turn to the more delicate contribution of τ ≥ t/2. For this case wewrite

(7.15) K(α)(t, τ) ≤ (1 + τ) supℓ 6=0

e−α|ℓ| supk

e−α|k(t−τ)+ℓτ |

1 + |k − ℓ|γ ,

and the upper bound is a nonincreasing function of d, so we assume d = 1. Withoutloss of generality we restrict the supremum to ℓ > 0.

The function x 7−→ (1+ |x− ℓ|γ)−1 e−α|x(t−τ)+ℓτ | is decreasing for x ≥ ℓ, increasingfor x ≤ −ℓτ/(t − τ); and on the interval [− ℓτ

t−τ, ℓ] its logarithmic derivative goes

from −α +

γℓt

1 +(

(t−τ)ℓt

)γ

(t− τ) to − α(t− τ).

So if t ≥ γ/α there is a unique maximum at x = −ℓτ/(t− τ), and the supremum in(7.15) is achieved for k equal to either the lower integer part, or the upper integerpart of −ℓτ/(t − τ). Thus a given integer k occurs in the supremum only for sometimes τ satisfying k − 1 < −ℓτ/(t − τ) < k + 1. Since only negative values of koccur, let us change the sign so that k is nonnegative. The equation

k − 1 <ℓτ

t− τ< k + 1

is equivalent to (k − 1

k + ℓ− 1

)t < τ <

(k + 1

k + ℓ+ 1

)t.


Moreover, τ > t/2 implies k ≥ ℓ. Thus, for t ≥ γ/α we have(7.16)

e−εt

∫ t

t/2

K(α)(t, τ) eετ dτ ≤ e−εt∑

ℓ≥1

e−αℓ∑

k≥ℓ

∫ ( k+1k+ℓ+1)t

( k−1k+ℓ−1)t

(1 + τ)e−α|k(t−τ)−ℓτ | eετ

1 + (k + ℓ)γdτ.

For t ≤ γ/α we have the trivial bound

e−εt

∫ t

t/2

K(α)(t, τ) eετ dτ ≤ γ

2α;

so in the sequel we shall just focus on the estimate of (7.16).

To evaluate the integral in the right-hand side of (7.16), we separate according towhether τ is smaller or larger than kt/(k + ℓ); we use trivial bounds for eετ insidethe integral, and in the end we get the explicit bounds

e−εt

∫ ( kk+ℓ)t

( k−1k+ℓ−1)t

(1 + τ) e−α|k(t−τ)−ℓτ | eετ dτ ≤ e−εℓtk+ℓ

[1

α(k + ℓ)+

kt

α(k + ℓ)2

],

e−εt

∫ ( k+1k+ℓ+1)t

( kk+ℓ)t

(1+τ) e−α|k(t−τ)−ℓτ | eετ dτ ≤ e−εℓt

k+ℓ+1

[1

α(k + ℓ)+

kt

α(k + ℓ)2+

1

α2(k + ℓ)2

].

All in all, there is a numeric constant C such that (7.16) is bounded above by

(7.17) C∑

ℓ≥1

e−αℓ∑

k≥ℓ

(1

α2(k + ℓ)2+γ+

1

α(k + ℓ)1+γ+

kt

α(k + ℓ)2+γ

)e−

ε ℓ tk+ℓ ,

together with an additional similar term where e−εℓt/(k+ℓ) is replaced by e−εℓt/(k+ℓ+1),and which will satisfy similar estimates.

We consider separately the three contributions in the right-hand side of (7.17).The first one is

1

α2

∑

ℓ≥1

e−αℓ∑

k≥ℓ

e−ε ℓtk+ℓ

(k + ℓ)2+γ.

To evaluate the behavior of this sum, we compare it to the two-dimensional integral

I(t) =1

α2

∫ ∞

1

e−αx

∫ ∞

x

e−ε xtx+y

(x+ y)2+γdy dx.


We change variables (x, y) → (x, u), where u(x, y) = εxt/(x+y). This has Jacobiandeterminant (dx dy)/(dx du) = (εxt)/u2, and we find

I(t) =1

α2 ε1+γ t1+γ

∫ ∞

1

e−αx

x1+γdx

∫ εt/2

0

e−u uγ du = O

(1

α2 ε1+γ t1+γ

).

The same computation for the second integral in the right-hand side of (7.17)yields

1

α εγ tγ

∫ ∞

1

e−αx

xγdx

∫ εt/2

0

e−u uγ−1 du = O

(ln 1

α

α εγ tγ

).

(The logarithmic factor arises only for γ = 1.)The third exponential in the right-hand side of (7.17) is the worse. It yields a

contribution

(7.18)t

α

∑

ℓ≥1

e−αℓ∑

k≥ℓ

e−ε ℓtk+ℓ k

(k + ℓ)2+γ.

We compare this with the integral

t

α

∫ ∞

1

e−αx

∫ ∞

x

e−ε xtx+y y

(x+ y)2+γdx dy,

and the same change of variables as before equates this with

1

α εγ tγ−1

∫ ∞

1

e−αx

xγdx

∫ εt/2

0

e−u uγ−1 du − 1

α ε1+γ tγ

∫ ∞

1

e−αx

xγdx

∫ εt/2

0

e−u uγ du

= O

(ln 1

α

α εγ tγ−1

).

(Again the logarithmic factor arises only for γ = 1.)The proof of Proposition 7.1 follows by collecting all these bounds and keeping

only the worse one.

Remark 7.3. It is not easy to catch (say numerically) the behavior of (7.18), be-cause it comes as a superposition of exponentially decaying modes; any truncationin k would lead to a radically different time-asymptotics.

From Proposition 7.1 we deduce L2 exponential bounds:

Corollary 7.4 (L2 exponential moments of the kernel). With the same notation asin Proposition 7.1,


0

2

4

6

8

10

12

14

16

0 5000 10000 15000 20000 25000

K = 0005K = 0010K = 0100K = 1000

0

2

4

6

8

10

12

14

16

0 20 40 60 80 100

K = 0005K = 0010K = 0100K = 1000

Figure 6. The function (7.18) for γ = 1, truncated at ℓ = 1 andk ≤ K, for K = 5, 10, 100, 1000. The decay is slower and slower, butstill exponential (picture on the left); however, the maximum valueoccurs on a much slower time scale and slowly increases with thetruncation parameter (picture on the right, which is a zoom on shortertimes).

(7.19) e−2εt

∫ t

0

K(α),γ(t, τ)2 e2ετ dτ ≤

C(γ)

α4 ε1+2γ t2(γ−1)if γ > 1

C

(1

α3 ε2+

1

α2 ε3 t

)if γ = 1.

Proof of Corollary 7.4. This follows easily from Proposition 7.1 and the obviousbound

K(α),γ(t, τ)2 ≤ C (1 + t)K(2α),2γ(t, τ).

7.3. Dual exponential moments.

Proposition 7.5. With the same notation as in Proposition 7.1, for any γ ≥ 1 wehave

(7.20) supτ≥0

eετ

∫ ∞

τ

e−εtK(α),γ(t, τ) dt ≤ C(γ)

(1

α2 ε+

ln 1α

α εγ

).


Remark 7.6. The corresponding computation for the baby model considered inSubsection 7.1.2 is

eετ

(1 + τ

α

) ∑

k≥1

e−ε( k+1k )τ

k1+γ≃(

1 + τ

α

)∫ ∞

1

e−ετ/x

x1+γdx

=

(1 + τ

τγ

) (1

α εγ

)∫ ετ

0

e−u uγ−1 du.

So we expect the dependence upon ε in (7.20) to be sharp for γ → 1.

Proof of Proposition 7.5. We first reduce to d = 1, and split the integral as

eετ

∫ ∞

τ

e−εtK(α),γ(t, τ) dt = eετ

∫ ∞

2τ

e−εtK(α),γ(t, τ) dt+ eετ

∫ 2τ

τ

e−εtK(α),γ(t, τ) dt

=: I1 + I2.

The first term I1 is easy: for 2τ ≤ t ≤ +∞ we have

K(α),γ(t, τ) ≤ (1 + τ)∑

k>1, ℓ 6=0

e−α|ℓ|−α2|k−ℓ| ≤ C (1 + τ)

α2,

and thus

eετ

∫ ∞

2τ

e−εtK(α),γ(t, τ) dt ≤ C (1 + τ)

α2e−ετ ≤ C

εα2.

We treat the second term I2 as in the proof of Proposition 7.1:

eετ

∫ 2τ

τ

K(α),γ(t, τ) e−εt dt

≤ eετ (1 + τ)∑

ℓ≥1

e−αℓ∑

k≥ℓ

∫ ( k+ℓ−1k−1 )τ

( k+ℓ+1k+1 )τ

e−α|k(t−τ)−ℓτ |

1 + (k + ℓ)γe−εt dt

≤ (1 + τ)∑

ℓ≥1

e−αℓ∑

k≥ℓ

e−ε ℓkτ

kγ

(2

kα

).


We compare this with

2 (1 + τ)

α

∫ ∞

1

e−αx

∫ ∞

x

e−ε xyτ

y1+γdy dx

=2

α εγ

(1 + τ

τγ

)∫ ∞

1

e−αx

xγ

∫ ετ

0

e−u uγ−1 du dx

≤ C ln(1/α)

α εγ,

where we used the change of variables u = εxτ/y. The desired conclusion follows.Note that as before the term ln(1/α) only occurs when γ = 1, and that for γ > 1,one could improve the estimate above into a time decay of the form O(τ−(γ−1)).

7.4. Growth control. To state the main result of this section we shall write Zd∗ =

Zd \ 0; and if a sequence of functions Φ(k, t) (k ∈ Zd∗, t ∈ R) is given, then

‖Φ(t)‖λ =∑

k e2πλ|k| |Φ(k, t)|. We shall useK(s) Φ(t) as a shorthand for (K(k, s) Φ(k, t))k∈Zd

∗,

etc.

Theorem 7.7 (Growth control via integral inequalities). Let f 0 = f 0(v) and W =W (x) satisfy condition (L) from Subsection 2.2 with constants C0, λ0, κ; in particular

|f 0(η)| ≤ C0 e−2πλ0|η|. Let further

CW = max

∑

k∈Zd∗

|W (k)|, supk∈Zd

∗

|k| |W (k)|

.

Let A ≥ 0, µ ≥ 0, λ ∈ (0, λ∗] with 0 < λ∗ < λ0. Let (Φ(k, t))k∈Zd∗, t≥0 be a continuous

function of t ≥ 0, valued in CZd∗ , such that

(7.21) ∀ t ≥ 0,

∥∥∥∥Φ(t) −∫ t

0

K0(t− τ) Φ(τ) dτ

∥∥∥∥λt+µ

≤ A +

∫ t

0

[K0(t, τ) +K1(t, τ) +

c0(1 + τ)m

]‖Φ(τ)‖λτ+µ dτ,

where c0 ≥ 0, m > 1 and K0(t, τ), K1(t, τ) are nonnegative kernels. Let ϕ(t) =‖Φ(t)‖λt+µ. Then

(i) Assume γ > 1 and K1 = cK(α),γ for some c > 0, α ∈ (0, α(γ)), where K(α),γ

is defined by (7.10), and α(γ) appears in Proposition 7.1. Then there are positive


constants C and χ, depending only on γ, λ∗, λ0, κ, c0, CW , m, uniform as γ → 1, suchthat if

(7.22) supt≥0

∫ t

0

K0(t, τ) dτ ≤ χ

and

(7.23) supt≥0

(∫ t

0

K0(t, τ)2 dτ

)1/2

+ supτ≥0

∫ ∞

τ

K0(t, τ) dt ≤ 1,

then for any ε ∈ (0, α),

(7.24) ∀ t ≥ 0, ϕ(t) ≤ C A(1 + c20)√

εeC c0

(1 +

c

α ε

)eCT eC c (1+T 2) eεt,

where

(7.25) T = C max

(c2

α5 ε2+γ

) 1γ−1

;

(c

α2 εγ+ 12

) 1γ−1

;

(c20ε

) 12m−1

.

(ii) Assume K1 =∑

1≤i≤N ciK(αi),1 for some αi ∈ (0, α(1)), where α(1) appears

in Proposition 7.1; then there is a numeric constant Γ > 0 such that whenever

1 ≥ ε ≥ ΓN∑

i=1

ciα3

i

,

one has, with the same notation as in (i),

(7.26) ∀ t ≥ 0, ϕ(t) ≤ C A(1 + c20) e

C c0

√ε

eCT eC c (1+T 2) eεt,

where

c =

N∑

i=1

ci, T = C max

1

ε2

(N∑

i=1

ciα3

i

);

(c20ε

) 12m−1

.

Remark 7.8. Let apart the term c0/(1 + τ)m which will appear as a technical cor-rection, there are three different kernels appearing in Theorem 7.7: the kernel K0,which is associated with the linearized Landau damping; the kernel K1, describingnonlinear echoes (due to interaction between differing Fourier modes); and the ker-nel K0, describing the instantaneous response (due to interaction between identicalFourier modes).


We shall first prove Theorem 7.7 assuming

(7.27) c0 = 0

and

(7.28)

∫ ∞

0

supk

|K0(k, t)| e2πλ0|k|t dt ≤ 1 − κ, κ ∈ (0, 1),

which is a reinforcement of condition (L). Under these assumptions the proof ofTheorem 7.7 is much simpler, and its conclusion can be substantially simplified too:χ depends only on κ; condition (7.23) on K0 can be dropped; and the factor eCT (1+

c/(α ε3/2)) in (7.24) can be omitted. If W ≤ 0 (as for gravitational interaction)

and f 0 ≥ 0 (as for Maxwellian background), these additional assumptions do notconstitute a loss of generality, since (7.28) becomes essentially equivalent to (L),and for c0 small enough the term c0(1 + τ)−m can be incorporated inside K0.

Proof of Theorem 7.7 under (7.23) and (7.28). We have

(7.29) ϕ(t) ≤ A +

∫ t

0

(|K0|(t− τ) +K0(t, τ) +K1(t, τ)

)ϕ(τ) dτ,

where |K0(t)| = supk |K0(k, t)|. We shall estimate ϕ by a maximum principleargument. Let ψ(t) = B eεt, where B will be chosen later. If ψ satisfies, for someT ≥ 0,

(7.30)

ϕ(t) < ψ(t) for 0 ≤ t ≤ T ,

ψ(t) ≥ A +

∫ t

0

(|K0|(t, τ) +K0(t, τ) +K1(t, τ)

)ψ(τ) dτ for t ≥ T ,

then u(t) := ψ(t) − ϕ(t) is positive for t ≤ T , and satisfies u(t) ≥∫ t

0K(t, τ) u(τ) dτ

for t ≥ T , with K = |K0| + K0 + K1 > 0; this prevents u from vanishing at latertimes, so u ≥ 0 and ϕ ≤ ψ. Thus it is sufficient to establish (7.30).

Case (i): By Proposition 7.1, and since∫

(|K0| +K0) dτ ≤ 1 − κ/2 (for χ ≤ κ/2),

(7.31) A+

∫ t

0

[|K0|(t, τ) +K0(t, τ)

]ψ(τ) dτ + c

∫ t

0

K(α),γ(t, τ)ψ(τ) dτ

≤ A +

[(1 − κ

2

)+

c C(γ)

α3 ε1+γ tγ−1

]B eεt.

For t ≥ T := (4 c C (α3 ε1+γκ))1/(γ−1), this is bounded above by A+ (1 − κ/4)B eεt,which in turn is bounded by B eεt as soon as B ≥ 4A/κ.


On the other hand, from the inequality

ϕ(t) ≤ A +(1 − κ

2

)sup

0≤τ≤tϕ(τ) + c (1 + t)

∫ t

0

ϕ(τ) dτ

we deduce

ϕ(t) ≤(

2A

κ

)(1 + t) e

2cκ

“t+ t2

2

”

In particular, if

4A

κec′(T+T 2) ≤ B

with c′ = c′(c, κ) large enough, then for 0 ≤ t ≤ T we have ϕ(t) ≤ ψ(t)/2, and (7.30)holds.

Case (ii): K1 =∑ciK

(αi),1. We use the same reasoning, replacing the right-handside in (7.31) by

A+

[(1 − κ

2

)+ C

(N∑

i=1

ciα3

i ε+

N∑

i=1

ciα3

i ε2 t

)]B eεt.

To conclude the proof, we may first impose a lower bound on ε to ensure

(7.32) C

N∑

i=1

ciα3

i ε≤ κ

8,

and then choose t large enough to guarantee

(7.33) CN∑

i=1

ciα3

i ε2 t

≤ κ

8;

this yields (ii).

Proof of Theorem 7.7 in the general case. We only treat (i), since the reasoning for(ii) is rather similar; and we only establish the conclusion as an a priori estimate,skipping the continuity/approximation argument needed to turn it into a rigorousestimate. Then the proof is done in three steps.


Step 1: Crude pointwise bounds. From (7.21) we have

ϕ(t) =∑

k∈Zd∗

|Φ(k, t)| e2π(λt+µ)|k|(7.34)

≤ A+∑

k

∫ t

0

∣∣K0(k, t− τ)∣∣ e2π(λt+µ)|k| |Φ(t, τ)| dτ

+

∫ t

0

[K0(t, τ) +K1(t, τ) +

c0(1 + τ)m

]ϕ(τ) dτ

≤ A+

∫ t

0

[(sup

k

∣∣K0(k, t− τ)∣∣ e2πλ(t−τ)|k|

)

+K1(t, τ) +K0(t, τ) +c0

(1 + τ)m

]ϕ(τ) dτ.

We note that for any k ∈ Zd∗ and t ≥ 0,

∣∣K0(k, t− τ)∣∣ e2πλ|k|(t−τ) ≤ 4π2 |W (k)|C0 e

−2π(λ0−λ)|k|t |k|2 t

≤ C C0

λ0 − λ

(supk 6=0

|k| |W (k)|)

≤ C C0CW

λ0 − λ,

where (here as below) C stands for a numeric constant which may change from lineto line. Assuming

∫K0(t, τ) dτ ≤ 1/2, we deduce from (7.34)

ϕ(t) ≤ A+1

2

(sup

0≤τ≤tϕ(τ)

)+ C

∫ t

0

(C0CW

λ0 − λ+ c (1 + t) +

c0(1 + τ)m

)ϕ(τ) dτ,

and by Gronwall’s lemma

(7.35) ϕ(t) ≤ 2AeC

“C0 CWλ0−λ

t+c(t+t2)+c0 Cm

”

,

where Cm =∫∞

0(1 + τ)−m dτ .

Step 2: L2 bound. This is the step where the smallness assumption (7.22) will bemost important. For all k ∈ Zd

∗, t ≥ 0, we define

(7.36) Ψk(t) = e−εt Φ(k, t) e2π(λt+µ)|k|,

(7.37) K0k(t) = e−εtK0(k, t) e2π(λt+µ)|k|,


Rk(t) = e−εt

(Φ(k, t) −

∫ t

0

K0(k, t− τ) Φ(k, τ) dτ

)e2π(λt+µ)|k|(7.38)

=(Ψk − Ψk ∗ K0

k

)(t),

and we extend all these functions by 0 for negative values of t. Taking Fouriertransform in the time variable yields Rk = (1 − K0

k) Ψk; since condition (L) implies

|1 − K0k| ≥ κ, we deduce ‖Ψk‖L2 ≤ κ−1 ‖Rk‖L2 , i.e.,

(7.39) ‖Ψk‖L2(dt) ≤‖Rk‖L2(dt)

κ.

Plugging (7.39) into (7.38), we deduce

(7.40) ∀ k ∈ Zd∗,

∥∥Ψk − Rk

∥∥L2(dt)

≤ ‖K0k‖L1(dt)

κ‖Rk‖L2(dt).

Then

∥∥ϕ(t) e−εt∥∥

L2(dt)=

∥∥∥∥∥∑

k

|Ψk|∥∥∥∥∥

L2(dt)

(7.41)

≤∥∥∥∥∥∑

k

|Rk|∥∥∥∥∥

L2(dt)

+∑

k

‖Rk − Ψk‖L2(dt)

≤∥∥∥∥∥∑

k

|Rk|∥∥∥∥∥

L2(dt)

1 +

1

κ

∑

ℓ∈Zd∗

‖K0ℓ‖L1(dt)

.

(Note: We bounded ‖Rℓ‖ by ‖∑k |Rk|‖, which seems very crude; but the decay ofK0

k as a function of k will save us.) Next, we note that

‖K0k‖L1(dt) ≤ 4π2 |W (k)|

∫ ∞

0

C0 e−2π(λ0−λ)|k|t |k|2 t dt

≤ 4π2 |W (k)| C0

(λ0 − λ)2,

so

∑

k

‖K0k‖L1(dt) ≤ 4π2

(∑

k

|W (k)|)

C0

(λ0 − λ)2.


Plugging this in (7.41) and using (7.21) again, we obtain

∥∥ϕ(t) e−εt∥∥

L2(dt)≤(

1 +C C0CW

κ (λ0 − λ)2

) ∥∥∥∥∥∑

k

|Rk|∥∥∥∥∥

L2(dt)

(7.42)

≤(

1 +C C0CW

κ (λ0 − λ)2

) ∫ ∞

0

e−2εt

(A +

∫ t

0

[K1 +K0 +

c0(1 + τ)m

]ϕ(τ) dτ

)2

dt

12

.

We separate this (by Minkowski’s inequality) into various contributions which weestimate separately. First, of course

(7.43)

(∫ ∞

0

e−2εtA2 dt

) 12

=A√2ε.

Next, for any T ≥ 1, by Step 1 and∫ t

0K1(t, τ) dτ ≤ Cc(1 + t)/α,

∫ T

0

e−2εt

(∫ t

0

K1(t, τ)ϕ(τ) dτ

)2

dt

12

(7.44)

≤[

sup0≤t≤T

ϕ(t)

](∫ T

0

e−2εt

(∫ t

0

K1(t, τ) dτ

)2

dt

) 12

≤ C AeC

hC0 CWλ0−λ

T+c (T+T 2)ic

α

(∫ ∞

0

e−2εt(1 + t)2 dt

) 12

≤ C Ac

α ε3/2e

Ch

C0 CWλ0−λ

T+c (T+T 2)i

.


Invoking Jensen and Fubini, we also have

∫ ∞

T

e−2εt

(∫ t

0

K1(t, τ)ϕ(τ) dτ

)2

dt

12

(7.45)

=

∫ ∞

T

(∫ t

0

K1(t, τ) e−ε(t−τ) e−ετ ϕ(τ) dτ

)2

dt

12

≤∫ ∞

T

(∫ t

0

K1(t, τ) e−ε(t−τ) dτ

)(∫ t

0

K1(t, τ) e−ε(t−τ) e−2ετϕ(τ)2 dτ

)dt

12

≤(

supt≥T

∫ t

0

e−εtK1(t, τ) eετ dτ

) 12(∫ ∞

T

∫ t

0

K1(t, τ) e−ε(t−τ) e−2ετϕ(τ)2 dτ dt

) 12

=

(supt≥T

∫ t

0


) 12(∫ ∞

0

∫ +∞

maxτ ; T

K1(t, τ) e−ε(t−τ) e−2ετ ϕ(τ)2 dt dτ

) 12

≤(

supt≥T

∫ t

0


) 12(

supτ≥0

∫ ∞

τ

eετ K1(t, τ) e−εt dt

) 12(∫ ∞

0

e−2ετ ϕ(τ)2 dτ

) 12

.

(Basically we copied the proof of Young’s inequality.) Similarly,

∫ ∞

0

e−2εt

(∫ t

0

K0(t, τ)ϕ(τ) dτ

)2

dt

12

(7.46)

≤(

supt≥0

∫ t

0


) 12(

supτ≥0

∫ ∞

τ


) 12(∫ ∞

0


) 12

≤(

supt≥0

∫ t

0

K0(t, τ) dτ

) 12(

supτ≥0

∫ ∞

τ

K0(t, τ) dt

) 12(∫ ∞

0


) 12

.


The last term is also split, this time according to τ ≤ T or τ > T :

∫ ∞

0

e−2εt

(∫ T

0

c0 ϕ(τ)

(1 + τ)mdτ

)2

dt

12

(7.47)

≤ c0

(sup

0≤τ≤Tϕ(τ)

)∫ ∞

0

e−2εt

(∫ T

0

dτ

(1 + τ)m

)2

dt

12

≤ c0C A√εe

Ch“

C0 CWλ0−λ

”T+c (T+T 2)

i

Cm,

and∫ ∞

0

e−2εt

(∫ t

T

c0 ϕ(τ) dτ

(1 + τ)m

)2

dt

12

(7.48)

= c0

∫ ∞

0

(∫ t

T

e−ε(t−τ) e−ετ ϕ(τ)

(1 + τ)mdτ

)2

dt

12

≤ c0

∫ ∞

0

(∫ t

T

e−2ε(t−τ)

(1 + τ)2mdτ

)(∫ t

T


)dt

12

≤ c0

(∫ ∞

0

e−2εt ϕ(t)2 dt

) 12(∫ ∞

0

∫ t

T

e−2ε(t−τ)

(1 + τ)2mdτ dt

) 12

= c0

(∫ ∞

0

e−2εt ϕ(t)2 dt

) 12(∫ ∞

T

1

(1 + τ)2m

(∫ ∞

τ

e−2ε(t−τ) dt

)dτ

) 12

= c0

(∫ ∞

0

e−2εt ϕ(t)2 dt

) 12(∫ ∞

T

dτ

(1 + τ)2m

) 12(∫ ∞

0

e−2εs ds

)12

=C

1/22m c0√

ε Tm−1/2

(∫ ∞

0

e−2εt ϕ(t)2 dt

) 12

.

Gathering estimates (7.43) to (7.48), we deduce from (7.42)

(7.49)∥∥ϕ(t) e−εt

∥∥L2(dt)

≤(

1 +C C0CW

κ (λ0 − λ)2

)C A√ε

[1 +

( c

α ε+ c0Cm

)]e

Ch

C0 CWλ0−λ

T+c (T+T 2)i

+ a∥∥ϕ(t) e−εt

∥∥L2(dt)

,


where

a =

(1 +

C C0CW

κ (λ0 − λ)2

) [(supt≥T

∫ t

0


) 12(

supτ≥0

∫ ∞

τ


) 12

+

(supt≥0

∫ t

0

K0(t, τ) dτ

) 12(

supτ≥0

∫ ∞

τ

K0(t, τ) dt

) 12

+C

1/22m c0√

ε Tm−1/2

].

Using Propositions 7.1 (case γ > 1) and 7.5, as well as assumptions (7.22) and(7.23), we see that a ≤ 1/2 for χ small enough and T satisfying (7.25). Then from(7.49) follows

∥∥ϕ(t) e−εt∥∥

L2(dt)≤(

1 +C C0CW

κ (λ0 − λ)2

)C A√ε

[1 +

( c

α ε+ c0Cm

)]e

Ch

C0 CWλ0−λ

T+c (T+T 2)i

.

Step 3: Refined pointwise bounds. Let us use (7.21) a third time, now for t ≥ T :

e−εt ϕ(t) ≤ Ae−εt +

∫ t

0

(sup

k|K0(k, t− τ)| e2πλ(t−τ)|k|

)ϕ(τ) e−ετ dτ

(7.50)

+

∫ t

0

[K0(t, τ) +

c0(1 + τ)m

]ϕ(τ) e−ετ dτ

+

∫ t

0

(e−εtK1(t, τ) e

ετ)ϕ(τ) e−ετ dτ

≤ Ae−εt +

[(∫ t

0

(supk∈Zd

∗

|K0(k, t− τ)| e2πλ(t−τ)|k|

)2

dτ

) 12

+

(∫ t

0

K0(t, τ)2 dτ

) 12

+

(∫ ∞

0

c20(1 + τ)2m

dτ

) 12

+

(∫ t

0

e−2εtK1(t, τ)2 e2ετ dτ

) 12] (∫ ∞

0

ϕ(τ)2 e−2ετ dτ

) 12

.


We note that, for any k ∈ Zd∗,(

|K0(k, t)| e2πλ|k|t)2

≤ 16 π4 |W (k)|2∣∣f 0(kt)

∣∣2 |k|4 t2 e4πλ|k|t

≤ C C20 |W (k)|2 e−4π(λ0−λ)|k|t |k|4 t2

≤ C C20

(λ0 − λ)2|W (k)|2 e−2π(λ0−λ)|k|t |k|2

≤ C C20

(λ0 − λ)2C2

W e−2π(λ0−λ)|k|t

≤ C C20

(λ0 − λ)2C2

W e−2π(λ0−λ)t;

so ∫ t

0

(supk∈Zd

∗

∣∣K0(k, t− τ)∣∣ e2πλ(t−τ)|k|

)2

dτ ≤ C C20 C

2W

(λ0 − λ)3.

Then the conclusion follows from (7.50), Corollary 7.4, conditions (7.25) and (7.23),and Step 2.

Remark 7.9. Theorem 7.7 leads to enormous constants, and it is legitimate toask about their sharpness, say with respect to the dependence in ε. We expect theconstant to be roughly of the order of

supt

(e(ct)

1/γ

e−εt)≃ exp

(ε−

1γ−1

).

Our bound is roughly like exp(ε−(4+2γ)/(γ−1)); this is worse, but displays the expectedbehavior as an exponential of an inverse power of ε, with a power that diverges likeO((1 − γ)−1) as γ → 1.

Remark 7.10. Even in the case of an analytic interaction, a similar argumentsuggests constants that are at best like (ln 1/ε)ln 1/ε, and this grows faster than anyinverse power of 1/ε.

To obtain sharper results, in Section 11 we shall later “break the norm” andwork directly on the Fourier modes of, say, the spatial density. In this subsection weestablish the estimates which will be used later; the reader who does not particularlycare about the case γ = 1 in Theorem 2.6 can skip them.

For any γ ≥ 1, α > 0, k, ℓ ∈ Zd \ 0 = Zd∗ and 0 ≤ τ ≤ t, we define

(7.51) K(α),γk,ℓ (t, τ) =

(1 + τ) d e−α|ℓ| e−α( t−τt )|k−ℓ| e−α|k(t−τ)+ℓτ |

1 + |k − ℓ|γ .


We start by exponential moment estimates.

Proposition 7.11. Let γ ∈ [1,∞) be given. For any α ∈ (0, 1), k, ℓ ∈ Zd∗, let K

(α),γk,ℓ

be defined by (7.51). Then there is α = α(γ) > 0 such that if α ≤ α and ε ∈ (0, α/4)then for any t > 0

(7.52) supk∈Zd

∗

∑

ℓ∈Zd∗

e−εt

∫ t

0

K(α),γk,ℓ (t, τ) eετ dτ ≤ C(d, γ)

α1+d εγ+1 tγ;

(7.53) supk∈Zd

∗

∑

ℓ∈Zd∗

e−εt

(∫ t

0

K(α),γk,ℓ (t, τ)2 e2ετ dτ

) 12

≤ C(d, γ)

αd εγ+ 12 tγ−

12

;

(7.54) supk∈Zd

∗

∑

ℓ∈Zd∗

supτ≥0

eετ

∫ ∞

τ

K(α),γk,ℓ e−εt dt ≤ C(d, γ)

α2+d ε.

Proof of Proposition 7.11. We first reduce to the case d = 1. Monotonicity cannotbe used now, but we note that

K(α),γk,ℓ (t, τ) ≤

∑

1≤j≤d

e−α|ℓ1| e−α|ℓ2| . . . e−α|ℓj−1|K(α),γkj ,ℓj

(t, τ) e−α|ℓj+1| . . . e−α|ℓd|,

where Kkj ,ℓjstands for a one-dimensional kernel. Thus

supk

∑

ℓ

∫ t

0

e−εtK(α),γk,ℓ (t, τ) eετ dτ ≤ sup

k

(∑

m∈Zd

e−α|m|

)d−1 ∑

1≤j≤d

∑

ℓj∈Z

∫ t

0

e−εtK(α),γkj ,ℓj

(t, τ) eετ dτ

≤ C(d)

αd−1sup

1≤j≤dsupkj∈Z

∑

ℓj∈Z

∫ t

0

e−εtK(α,γ)kj ,ℓj

(t, τ) eετ dτ.

In other words, for (7.52) we may just consider the one-dimensional case, providedwe allow an extra multiplicative constant C(d)/αd−1. A similar reasoning holds for(7.53) and (7.54). From now on we focus on the case d = 1.

Without loss of generality we assume k > 0, and only treat the worse case ℓ < 0.(The other case k, ℓ > 0 is simpler and yields an exponential decay in time of the

form e−c minα,εt). For simplicity we also write Kk,ℓ = K(α),γk,ℓ . An easy computation


yields

e−εt

∫ t

0

Kk,ℓ(t, τ) eετ dτ

≤ C e−α|ℓ|

1 + |k − ℓ|γ(

1

α|k − ℓ| +|k|t

α|k − ℓ|2 +1

α2|k − ℓ|2)e−

ε|ℓ|t|k−ℓ| .

Then for any k ≥ 1, we have (crudely writing α2 = O(α))

(7.55)∑

ℓ≤−1

∫ t

0

e−εtKk,ℓ(t, τ) eετ dτ

≤ C

(∑

ℓ≥1

e−αℓ e−εℓtk+ℓ

α2 (k + ℓ)1+γ+∑

ℓ≥1


α (k + ℓ)2+γkt

).

For the first sum in the right-hand side of (7.55) we write

∑

ℓ≥1


(k + ℓ)1+γ≤∑

ℓ≥1

e−αℓ

ℓ1+γ

[(εℓt

k + ℓ

)1+γ

e−εℓtk+ℓ

]1

(εt)1+γ(7.56)

≤ C(γ)

(εt)1+γ.

For the second sum in the right-hand side of (7.55) we separate according to1 ≤ ℓ ≤ k or ℓ ≥ k + 1:

∑

1≤ℓ≤k


(k + ℓ)2+γkt ≤

∑

1≤ℓ≤k

e−αℓ e−εt

k+1

(k + 1)2+γkt(7.57)

≤ C

α

[e−

εtk+1

(εt

k + 1

)1+γ](

k

k + 1

)t

(εt)1+γ

≤ C

α ε1+γ tγ;

∑

ℓ≥k+1


(k + ℓ)2+γkt ≤ C

∑

ℓ≥k+1

e−αℓ e−εt2

k2+γkt(7.58)

≤ C

α

e−εt/4

ε k1+γ≤ C

α ε1+γ tγ.

The combination of (7.55) (7.56), (7.57) and (7.58) completes the proof of (7.52).


Now we turn to (7.53). The estimates are rather similar, since

Kk,ℓ(t, τ)2 ≤ C (1 + t)Kk,ℓ(t, τ)

with γ → 2γ and α→ 2α. So (7.55) should be replaced by

(7.59)∑

ℓ

e−εt

(∫ t

0

Kk,ℓ(t, τ)2 e2ετ dτ

) 12

≤ C

(∑

ℓ≥1

e−αℓ e−εℓtk+ℓ (1 + t)1/2

α (k + ℓ)12+γ

+∑

ℓ≥1

e−αℓ e−εℓtk+ℓ (kt)1/2 (1 + t)1/2

α1/2 (k + ℓ)1+γ

).

For the first sum we use (7.56) with γ replaced by γ − 1/2: for t ≥ 1,

(7.60) (1 + t)1/2∑

ℓ


(k + ℓ)1+(γ−1/2)≤ C(γ) t1/2

(εt)γ+1/2≤ C(γ)

εγ+1/2tγ.

For the second sum in the right-hand side of (7.59) we write

∑

1≤ℓ≤k

e−αℓ e−εℓtk+ℓ k1/2t

(k + ℓ)1+γ≤ C

∑

ℓ

e−αℓ

[e−

εtk+1

(εt

k + 1

)γ+1/2]

k1/2

(1 + k)1/2

t

(εt)γ+1/2

≤ C

α εγ+ 12 tγ−

12

and∑

ℓ≥k+1

e−αℓ e−εℓtk+ℓ k1/2t

(k + ℓ)1+γ≤ C

∑

ℓ

e−αℓ

ℓγ+ 12

e−εt2 t ≤ C e−

εtt ≤ C

(ε t)γ− 12 ε.

With this (7.53) is readily obtained.

Finally we consider (7.54). As in Proposition 7.5 one easily shows that

supk

∑

ℓ

supτeετ

∫ ∞

2τ

e−εtKk,ℓ(t, τ) dτ ≤ C

εα2

∑

ℓ

e−α|ℓ| ≤ C

εα3.

Then one has

eετ

∫ 2τ

τ

e−α|k(t−τ)+ℓτ | e−εt dt ≤ C

α2k+

C

αεk+

C

αke−

εℓτk .


So the problem amounts to estimate

∑

ℓ

supτ

[(1 + τ)

e−αℓ e−εℓτk

αk(k + ℓ)γ

]≤∑

ℓ

e−αℓ

[1

α+

1

εℓ(k + ℓ)γ

(e−

εℓτkεℓτ

k

)]

≤ C

(1

α2+

1

ε

),

and the proof is complete.

We conclude this section with a mode-by-mode analogue of Theorem 7.7.

Theorem 7.12. Let f 0 = f 0(v) and W = W (x) satisfy condition (L) from Sub-

section 2.2 with constants C0, λ0, κ; in particular |f 0(η)| ≤ C0 e−2πλ0|η|. Further

let

CW = max

∑

k∈Zd∗

|W (k)|, supk∈Zd

∗

|k| |W (k)|

.

Let (Ak)k∈Zd∗, µ ≥ 0, λ ∈ (0, λ∗] with 0 < λ∗ < λ0. Let (Φ(k, t))k∈Zd

∗, t≥0 be a

continuous function of t ≥ 0, valued in CZd∗ , such that for all t ≥ 0 and k ∈ Zd

∗,

(7.61)

e2π(λt+µ)|k|∣∣∣Φ(k, t)−

∫ t

0

K0(k, t−τ) Φ(k, τ) dτ∣∣∣ ≤ Ak+

∫ t

0

K0(t, τ) e2π(λτ+µ)|k| |Φ(k, τ)| dτ

+

∫ t

0

∑

ℓ∈Zd∗

(cK

(α),γk,ℓ (t, τ) +

cℓ(1 + τ)m

)e2π(λτ+µ)|k−ℓ| |Φ(k − ℓ, τ)| dτ,

where c > 0, cℓ ≥ 0 (ℓ ∈ Zd∗), m > 1, γ ≥ 1, K0(t, τ) is a nonnegative kernel, K

(α),γk,ℓ

are defined by (7.51), α < α(γ) defined in Proposition 7.11. Then there are positive

constants C and χ, depending only on γ, λ∗, λ0, κ, c := max∑

ℓ cℓ, (∑

ℓ c2ℓ)

1/2,CW , m, such that if

(7.62) supt≥0

∫ t

0

K0(t, τ) dτ ≤ χ

and

(7.63) supt≥0

(∫ t

0

K0(t, τ)2 dτ

)1/2

+ supτ≥0

∫ ∞

τ

K0(t, τ) dt ≤ 1,


then for any ε ∈ (0, α/4) and for any t ≥ 0,

(7.64) supk

(e2π(λt+µ)|k| |Φ(k, t)|

)≤ C A

(1 + c2)√ε

eC c(1 +

c

α2 ε

)eCT eC c

α(1+T 2) eεt,

where A := (supk Ak) and

(7.65) T = C max

(c2

α3+2d εγ+2

) 1γ

;

(c

αd εγ+ 12

) 1

γ− 12

;

(c2

ε

) 12m−1

.

Proof of Theorem 7.12. The proof is quite similar to the proof of Theorem 7.7, sowe shall only point out the differences. As in the proof of Theorem 7.7 we startby crude pointwise bounds obtained by Gronwall inequality; but this time on thequantity

ϕ(t) = supk

|Φ(k, t)| e2π(λt+µ)|k|.

Since∑

ℓKk,ℓ(t, τ) = O((1 + τ)/α), we find

(7.66) ϕ(t) ≤ 2 A eC

“C0 CWλ0−λ

t+ cα

(t+t2)+c Cm

”

.

Next we define Ψk, K0k, Rk as in Step 2 of the proof of Theorem 7.7, and we deduce

(7.39) and (7.40). Let

(7.67) ϕk(t) = e2π(λt+µ)|k| |Φ(k, t)|,then

∥∥ϕk(t) e−εt∥∥

L2(dt)≤ ‖Rk‖L2(dt)

(1 +

‖K0k‖L1(dt)

κ

)

≤ ‖Rk‖L2(dt)

(1 +

C CW C0

κ

);

whence

(7.68)∥∥ϕk(t) e

−εt∥∥

L2(dt)≤(

1 +C C0CW

κ (λ0 − λ)2

) ∫ ∞

0

e−2εt

(Ak +

∫ t

0

K0(t, τ)ϕk(τ) dτ

+∑

ℓ

∫ t

0

(cKk,ℓ(t, τ) +

cℓ(1 + τ)m

)ϕk−ℓ(τ) dτ

)2

dt

12

.


We separate this into various contributions as in the proof of Theorem 7.7. Inparticular, using (7.66) and

∫ t

0

∑ℓKk,ℓ dτ = O((1 + t)/α2), we find

∫ T

0

e−2εt

(∫ t

0

∑

ℓ

Kk,ℓ(t, τ)ϕk−ℓ(τ) dτ

)2

dt

12

(7.69)

≤[sup

ksup

0≤t≤Tϕk(t)

]∫ T

0

e−2εt

(∫ t

0

∑

ℓ

Kk,ℓ(t, τ) dτ

)2

dt

12

≤ C Ac

α2 ε3/2e

Ch

C0 CWλ0−λ

T+ cα

(T+T 2)i

.

Also,

∫ ∞

T

e−2εt

(∫ t

0

∑

ℓ

Kk,ℓ(t, τ)ϕk−ℓ(τ) dτ

)2

dt

12

≤(

supt≥T

∫ t

0

e−εt∑

ℓ

Kk,ℓ(t, τ) eετ dτ

) 12(∫ ∞

T

∫ t

0

∑

ℓ

Kk,ℓ(t, τ) e−ε(t−τ) e−2ετϕk−ℓ(τ)

2 dτ dt

) 12

,

and the last term inside parentheses is

∑

ℓ

∫ ∞

0

(∫ ∞

maxτ ;T

Kk,ℓ(t, τ) e−ε(t−τ) dt

)e−2ετ ϕk−ℓ(τ)

2 dτ

≤(∑

ℓ

supτ

∫ ∞

τ

Kk,ℓ(t, τ) e−ε(t−τ) dt

) [sup

ℓ

∫e−2ετ ϕℓ(τ)

2 dτ

].

The computation for K0 is the same as in the proof of Theorem 7.7, and the termsin (1 + τ)−m are handled in essentially the same way: simple computations yield

∫ ∞

0

e−2εt

(∫ T

0

∑ℓ cℓ ϕk−ℓ(τ)

(1 + τ)mdτ

)2

dt

12

(7.70)

≤(

sup0≤τ≤T

supℓϕℓ(τ)

)∫ ∞

0

e−2εt

(∫ T

0

(∑cℓ) dτ

(1 + τ)m

)2

dt

12

≤ cCm A√

εe

Ch“

C0 CWλ0−λ

”T+ c

α(T+T 2)

i


and

∫ ∞

0

e−2εt

(∫ t

T

∑ℓ cℓ ϕk−ℓ(τ) dτ

(1 + τ)m

)2

dt

12

(7.71)

≤

sup

t≥0, ℓ

(∫ t

T

e−2ετ ϕℓ(τ)2 dτ

) (∑

ℓ

cℓ

)2(∫ ∞

0

∫ t

T

e−2ε(t−τ)

(1 + τ)2mdτ dt

)

12

≤ c

(C2m

ε T 2m−1

) 12(

supℓ

∫ +∞

0

e−2ετ ϕℓ(τ)2 dτ

) 12

.

All in all, we end up with

(7.72)

supk


L2(dt)≤(

1 +C C0CW

κ (λ0 − λ)2

)C A√ε

[1 +

( c

α2 ε+ c Cm

)]e

Ch

C0 CWλ0−λ

T+ cα

(T+T 2)i

+ a supk


L2(dt),

where

a =

(1 +

C C0CW

κ (λ0 − λ)2

) [

c2

(supt≥T

∑

ℓ

∫ t

0

e−εtKk,ℓ(t, τ) eετ dτ

) 12(∑

ℓ

supτ≥0

∫ ∞

τ

eετ Kk,ℓ(t, τ) e−εt dt

) 12

+

(supt≥0

∫ t

0

K0(t, τ) dτ

) 12(

supτ≥0

∫ ∞

τ

K0(t, τ) dt

) 12

+C

1/22m c0√

ε Tm−1/2

].

Applying Proposition 7.11, we see that a ≤ 1/2 as soon as T satisfies (7.65), andthen we deduce from (7.72) a bound on supk ‖ϕk(t) e

−εt‖L2(dt).


Finally, we conclude as in Step 3 of the proof of Theorem 7.7: from (7.61),

e−εt ϕk(t) ≤ Ak e−εt +

[(∫ t

0

(supk∈Zd

∗

|K0(k, t− τ)| e2πλ(t−τ)|k|

)2

dτ

) 12

(7.73)

+

(∫ t

0

K0(t, τ)2 dτ

) 12

+ c

(∫ ∞

0

dτ

(1 + τ)2m

) 12

+ c∑

ℓ

(∫ t

0

e−2εtKk,ℓ(t, τ)2 e2ετ dτ

) 12

] (sup

k

∫ ∞

0

ϕk(τ)2 e−2ετ dτ

) 12

,

and the conclusion follows by a new application of Proposition 7.11.

8. Approximation schemes

Having defined a functional setting (Section 4) and identified several mathemat-ical/physical mechanisms (Sections 5 to 7), we are prepared to fight the Landaudamping problem. For that we need an approximation scheme solving the nonlinearVlasov equation. The problem is not to prove the existence of solutions (this is mucheasier), but to devise the scheme in such a way that it leads to relevant estimatesfor our study.

The first idea which may come to mind is a classical Picard scheme for quasilinearequations:

(8.1) ∂tfn+1 + v · ∇xf

n+1 + F [fn] · ∇vfn+1 = 0.

This has two drawbacks: first, fn+1 evolves by the characteristics created by F [fn],and this will deteriorate the estimates in analytic regularity. Secondly, there is nohope to get a closed (or approximately closed) equation on the density associatedwith fn+1. More promising, and more in the spirit of the linearized approach, wouldbe a scheme like

(8.2) ∂tfn+1 + v · ∇xf

n+1 + F [fn+1] · ∇vfn = 0.

(Physically, fn+1 forces fn, and the question is whether the reaction will exhaustfn+1 in large time.) But when we write (8.2) we are implicitly treating a higherorder term (∇vf) of the equation in a perturbative way; so this has no reason toconverge.

To circumvent these difficulties, we shall use a Newton iteration: not only willthis provide more flexibility in the regularity indices, but at the same time it willyield an extremely fast rate of convergence (something like O(ε2n

)) which will be


most welcome to absorb the large constants coming from Theorem 7.7 or Theorem7.12.

8.1. The natural Newton scheme. Let us adapt the abstract Newton scheme toan abstract evolution equation in the form

∂f

∂t= Q(f),

around a stationary solution f 0 (so Q(f 0) = 0). Write the Cauchy problem withinitial datum fi ≃ f 0 in the form

Φ(f) :=(∂tf −Q(f), f(0, · )

)− (0, fi).

Starting from f 0, the Newton iteration consists in solving inductively Φ(fn−1) +Φ′(fn−1) · (fn − fn−1) = 0 for n ≥ 1. More explicitly, writing hn = fn − fn−1, weshould solve

∂th1 = Q′(f 0) · h1

h1(0, · ) = fi − f 0

∀n ≥ 1,

∂th

n+1 = Q′(fn) · hn+1 −[∂tf

n −Q(fn)]

hn+1(0, · ) = 0.

By induction, for n ≥ 1 this is the same as∂th

n+1 = Q′(fn) · hn+1 +[Q(fn−1 + hn) −Q(fn−1) −Q′(fn−1) · hn

]

hn+1(0, · ) = 0.

This is easily applied to the nonlinear Vlasov equation, for which the nonlinear-ity is quadratic. So we define the natural Newton scheme for the nonlinearVlasov equation as follows:

f 0 = f 0(v) is given (homogeneous stationary state)

fn = f 0 + h1 + . . .+ hn, where

(8.3)

∂th

1 + v · ∇xh1 + F [h1] · ∇vf

0 = 0

h1(0, · ) = fi − f 0


(8.4)

∀n ≥ 1,

∂th

n+1 + v · ∇xhn+1 + F [fn] · ∇vh

n+1 + F [hn+1] · ∇vfn = −F [hn] · ∇vh

n

hn+1(0, · ) = 0.

Here F [f ] is the force field created by the particle distribution f , namely

(8.5) F [f ](t, x) = −∫∫

Td×Rd

∇W (x− y) f(t, y, w) dy dw.

Note also that all the ρn =∫hn dv for n ≥ 1 have zero spatial average.

8.2. Battle plan. The treatment of (8.3) was performed in Subsection 4.12. Nowthe problem is to handle all equations appearing in (8.4). This is much more com-plicated, because for n ≥ 1 the background density fn depends on t and x, insteadof just v; as a consequence,

(a) Equation (8.4) cannot be considered as a perturbation of free transport, be-cause of the presence of ∇vh

n+1 in the left-hand side;

(b) The reaction term F [hn+1] · ∇vfn no longer has the simple product structure

(function of x)×(function of v), so it becomes harder to get hands on the homoge-nization phenomenon;

(c) Because of spatial inhomogeneities, echoes will appear; they are all the moredangerous that, ∇vf

n is unbounded as t→ ∞, even in gliding regularity. (It growslike O(t), which is reminiscent of the observation made by Backus [4].)

The estimates in Sections 5 to 7 have been designed precisely to overcome theseproblems; however we still have a few conceptual difficulties to solve before applyingthese tools.

Recall the discussion in Subsection 4.11: the natural strategy is to propagate thebound

(8.6) supτ≥0

‖fτ‖Zλ,µ;1τ

< +∞

along the scheme; this estimate contains in particular two crucial pieces of informa-tion:• a control of ρτ =

∫fτ dv in Fλτ+µ norm;

• a control of 〈fτ 〉 =∫fτ dx in Cλ;1 norm.

So the plan would be to try to get inductively estimates of each hn in a norm likethe one in (8.6), in such a way that hn is extremely small as n→ ∞, and allowing a


slight deterioration of the indices λ, µ as n → ∞. Let us try to see how this wouldwork: assuming

∀ 0 ≤ k ≤ n, supτ≥0

‖hkτ‖Zλk,µk ;1

τ≤ δk,

we should try to bound hn+1τ . To “solve” (8.4), we apply the classical method of

characteristics: as in Section 5 we define (Xnτ,t, V

nτ,t) as the solution of

d

dtXn

τ,t(x, v) = V nτ,t(x, v),

d

dtV n

τ,t(x, v) = F [fn](t, Xn

τ,t(x, v))

Xnτ,τ(x, v) = x, V n

τ,τ (x, v) = v.

Then (8.4) is equivalent to

(8.7)d

dthn+1

(t, Xn

0,t, Vn0,t(x, v)

)= Σn+1

(t, Xn

0,τ(x, v), Vn0,τ (x, v)

),

where

(8.8) Σn+1(t, x, v) = −F [hn+1] · ∇vfn − F [hn] · ∇vh

n.

Integrating (8.7) in time and recalling that hn+1(0, ·) = 0, we get

hn+1(t, Xn

0,t(x, v), Vn0,t(x, v)

)=

∫ t

0

Σn+1(τ,Xn

0,τ (x, v), Vn0,τ(x, v)

)dτ.

Composing with (Xnt,0, V

nt,0) and using (5.2) yields

hn+1(t, x, v) =

∫ t

0

Σn+1(τ,Xn

t,τ (x, v), Vnt,τ (x, v)

)dτ.

We rewrite this using the deflection map

Ωnt,τ (x, v) = (Xn

t,τ , Vnt,τ )(x+ v(t− τ), v) = Sn

t,τ S0τ,t;

then we finally obtain

hn+1(t, x, v) =

∫ t

0

(Σn+1

τ Ωnt,τ

)(x− v(t− τ), v) dτ(8.9)

= −∫ t

0

[(F [hn+1

τ ] Ωnt,τ

)·((

∇vfnτ

) Ωn

t,τ

)](x− v(t− τ), v) dτ

−∫ t

0

[(F [hn

τ ] Ωnt,τ

)·((

∇vhnτ

) Ωn

t,τ

)](x− v(t− τ), v) dτ.


Since the unknown hn+1 appears on both sides of (8.9), we need to get a self-consistent estimate. For this we have little choice but to integrate in v and get anintegral equation on ρ[hn+1] =

∫hn dv, namely

(8.10) ρ[hn+1](t, x) =

∫ t

0

∫ [((ρ[hn+1

τ ] ∗ ∇W) Ωn

t,τ

)·Gn

τ,t

] S0

τ−t(x, v) dv dτ

+ (stuff from stage n),

where Gnτ,t = ∇vf

nτ Ωn

t,τ . By induction hypothesis Gnτ,t is smooth with regularity

indices roughly equal to λn, µn; so if we accept to lose just a bit more on the regu-larity we may hope to apply the long-term regularity extortion and decay estimatesfrom Section 6, and then time-response estimates of Section 7, and get the desireddamping.

However, we are facing a major problem: composition of ρ[hn+1τ ] ∗ ∇W by Ωn

t,τ

implies a loss of regularity in the right-hand side with respect to the left-hand side,which is of course unacceptable if one wants a closed estimate. The short-termregularity extortion from Section 6 remedies this, but the price to pay is that Gn

should now be estimated at time τ ′ = τ − bt/(1 + b) instead of τ , and with index ofgliding analytic regularity roughly equal to λn(1+ b) rather than λn. Now the catchis that the error induced by composition by Ωn depends on the whole distributionfn, not just hn; thus, if the parameter b should control this error it should stay oforder 1 as n→ ∞, instead of converging to 0.

So it seems we are sentenced to lose a fixed amount of regularity (or rather of ra-dius of convergence) in the transition from stage n to stage n+1; this is reminiscentof the “Nash–Moser syndrom” [2]. The strategy introduced by Nash [68] to remedysuch a problem (in his case arising in the construction of C∞ isometric imbed-dings) consisted in combining a Newton scheme with regularization; his method waslater developed by Moser [65] for the C∞ KAM theorem (see [66, pp. 19–21] forsome interesting historical comments). A clear and concise proof of the Nash–Moserimplicit function theorem, together with its application to the C∞ embedding prob-lem, can be found in [81]. The Nash–Moser method is arguably the most powerfulperturbation technique known to this day. However, despite significant effort, wewere unable to set up any relevant regularization procedure (in gliding regularity, ofcourse) which could be used in Nash–Moser style, because of three serious problems:

• The convergence of the Nash–Moser scheme is no longer as fast as that of the“raw” Newton iteration; instead, it is determined by the regularity of the data, and


the resulting rates would be unlikely to be fast enough to win over the giganticconstants coming from Section 7.

• Analytic regularization in the v variable is extremely costly, especially if wewish to keep a good localization in velocity space, as the one appearing in Theo-rem 4.20(iii), that is exponential integrability in v; then the uncertainty principlebasically forces us to pay O(eC/ε2

), where ε is the strength of the regularization.

• Regularization comes with an increase of amplitude (there is as usual a trade-off between size and regularity); if we regularize before composition by Ωn, this willdevastate the estimates, because the analytic regularity of f g depends not onlyon the regularity of f and g, but also on the amplitude of g − Id .

Fortunately, it turned out that a “raw” Newton scheme could be used; but thisrequired to give up the natural estimate (8.6), and replace it by the pair of estimates

(8.11)

supτ≥0

‖ρτ‖Fλτ+µ < +∞;

supt≥τ≥0

∥∥∥fτ Ωt,τ

∥∥∥Z

λ(1+b),µ;1

τ− bt1+b

< +∞.

Here b = b(t) takes the form const./(1 + t), and is kept fixed all along the scheme;moreover λ, µ will be slightly larger than λ, µ, so that none of the two estimatesin (8.11) implies the other one. Note carefully that there are now two times (t, τ)explicitly involved, so this is much more complex than (8.6). Let us explain whythis strategy is nonetheless workable.

First, the density ρn =∫fn dv determines the characteristics at stage n, and

therefore the associated deflection Ωn. If ρnτ is bounded in Fλnτ+µ, then by Theorem

5.2 we can estimate Ωnt,τ in Zλ′

n,µ′n

τ ′ , as soon as (essentially) λ′n τ′ + µ′

n ≤ λn τ + µn,λ′n < λn, and these bounds are uniform in t.

Of course, we cannot apply this theorem in the present context, because λn(1+b) isnot bounded above by λn. However, for large times t we may afford λn(1+b(t)) < λn,while λn(1 + b)(τ − bt/(1 + b)) ≤ λnτ for all times; this will be sufficient to repeatthe arguments in Section 5, getting uniform estimates in a regularity which dependson t. (The constants are uniform in t; but the index of regularity goes down witht.) We can also do this while preserving the other good properties from Theorem5.2, namely exponential decay in τ , and vanishing near τ = t.

Figure 7 below summarizes schematically the way we choose and estimate thegliding regularity indices.


.

t

λn

λn(1 + b)

λ∞ = λ∞

λn+1

λn+1(1 + b)..

Figure 7. Indices of gliding regularity appearing throughout ourNewton scheme, respectively in the norm of ρ[hτ ] and in the normof hτ Ωt,τ , plotted as functions of t

Besides being uniform in t, our bounds need to be uniform in n. For this we shallhave to stratify all our estimates, that is decompose ρ[fn] = ρ[h1] + · · ·+ ρ[hn], andconsider separately the influence of each term in the equations for characteristics.This can work only if the scheme converges very fast.

Once we have estimates on Ωnt,τ in a time-varying regularity, we can work with

the kinetic equation to derive estimates on hnτ Ωn

t,τ ; and then on all hkτ Ωn

t,τ , alsoin a norm of time-varying regularity. We can also estimate their spatial average,

in a norm Cλ(1+b);1; thanks to the exponential convergence of the deflection map asτ → ∞ these estimates will turn out to be uniform in τ .

Next, we can use all this information, in conjunction with Theorem 6.5, to getan integral inequality on the norm of ρ[hn+1

τ ] in Fλτ+µ, where λ and µ are onlyslightly smaller than λn and µn. Then we can go through the response estimatesof Section 7, this gives us an arbitrarily small loss in the exponential decay rate,at the price of a huge constant which will eventually be wiped out by the fastconvergence of the scheme. So we have an estimate on ρ[hn+1], and we are inbusiness to continue the iteration. (To ensure the propagation of the linear dampingcondition, or equivalently of the smallness of K0 in Theorem 7.7, throughout thescheme, we shall have to stratify the estimates once more.)


9. Local in time iteration

Before working out the core of the proof of Theorem 2.6 in Section 10, we shallneed a short-time estimate, which will act as an “initial regularity layer” for theNewton scheme. (This will give us room later to allow the regularity index todepend on t.) So we run the whole scheme once in this section, and another time inthe next section.

Short-time estimates in the analytic class are not new for the nonlinear Vlasovequation: see in particular the work of Benachour [8] on Vlasov–Poisson. His ar-guments can probably be adapted for our purpose; also the Cauchy–Kowalevskayamethod could certainly be applied. We shall provide here an alternative method,based on the analytic function spaces from Section 4, but not needing the apparatusfrom Sections 5 to 7. Unlike the more sophisticated estimates which will be per-formed in Section 10, these ones are “almost” Eulerian (the only characteristics arethose of free transport). The main tool is the

Lemma 9.1. Let f be an analytic function, λ(t) = λ − K t, µ(t) = µ − K t; letT > 0 be so small that λ(t), µ(t) > 0 for 0 ≤ t ≤ T . Then for any τ ∈ [0, T ] andany p ≥ 1,

(9.1)d

dt

+∣∣∣∣t=τ

‖f‖Z

λ(t),µ(t);pτ

≤ − K

1 + τ‖∇f‖

Zλ(τ),µ(τ);pτ

,

where (d+/dt) stands for the upper right derivative.

Remark 9.2. Time-differentiating Lebesgue integrability exponents is common prac-tice in certain areas of analysis; see e.g. [33]. Time-differentiation with respect toregularity exponents is less common; however, as pointed out to us by Strain, Lemma9.1 is strongly reminiscent of a method recently used by Chemin [17] to derive localanalytic regularity bounds for the Navier–Stokes equation. We expect that simi-lar ideas can be applied to more general situations of Cauchy–Kowalevskaya type,especially for first-order equations, and maybe this has already been done.


Proof of Lemma 9.1. For notational simplicity, let us assume d = 1. The left-handside of (9.1) is

∑

n,k

e2πµ(τ)|k| 2πµ(τ) |k| λn(τ)

n!

∥∥∥(∇v + 2iπkτ

)nf(k, v)

∥∥∥Lp(dv)

+∑

n,k

e2πµ(τ)|k| λ(τ)λn−1(τ)

(n− 1)!

∥∥∥(∇v + 2iπkτ

)nf(k, v)

∥∥∥Lp(dv)

≤ −K∑

n,k

e2πµ(τ)|k| 2π |k| λn(τ)

n!

∥∥∥(∇v + 2iπkτ

)nf(k, v)

∥∥∥Lp(dv)

−K∑

n,k

e2πµ(τ)|k| λn(τ)

n!

∥∥∥(∇v + 2iπkτ

)n+1f(k, v)

∥∥∥Lp(dv)

≤ −K∑

n,k


n!

∥∥∥(∇v + 2iπkτ

)n∇xf(k, v)∥∥∥

Lp(dv)

+Kτ

1 + τ

∑

n,k


n!

∥∥∥(∇v + 2iπkτ

)n∇xf(k, v)∥∥∥

Lp(dv)

− K

1 + τ

∑

n,k


n!

∥∥∥(∇v + 2iπkτ

)n∇vf(k, v)∥∥∥

Lp(dv),

where in the last step we used ‖(∇v + 2iπkτ)h‖ ≥ (1/(1 + τ))(‖∇vh‖ − τ‖2iπkh‖).The conclusion follows.

Now let us see how to propagate estimates through the Newton scheme describedin Section 10. The first stage of the iteration (h1 in the notation of (8.3)) wasconsidered in Subsection 4.12, so we only need to care about higher orders. For any

k ≥ 1 we solve ∂thk+1 + v · ∇xh

k+1 = Σk+1, where

Σk+1 = −(F [hk+1] · ∇vf

k + F [fk] · ∇vhk+1 + F [hk] · ∇vh

k)

(note the difference with (8.7)–(8.8)). Recall that fk = f 0 + h1 + . . . + hk. Wedefine λk(t) = λk − 2K t, µk(t) = µk −K t, where (λk)k∈N, (µk)k∈N are decreasingsequences of positive numbers.

We assume inductively that at stage n of the iteration, we have constructed(λk)k≤n, (µk)k≤n, (δk)k≤n such that

∀ k ≤ n, sup0≤t≤T

∥∥hk(t, · )∥∥Z

λk(t),µk(t);1t

≤ δk,


for some fixed T > 0. The issue is to construct λn+1, µn+1 and δn+1 so that theinduction hypothesis is satisfied at stage n+ 1.

At t = 0, hn+1 = 0. Then we estimate the time-derivative of ‖hn+1‖Z

λn+1(t),µn+1(t);1

t

.

Let us first pretend that the regularity indices λn+1 and µn+1 do not depend on t;

then hn+1(t) =∫ t

0Σn+1 S0

−(t−τ) dτ , so by Proposition 4.19

‖hn+1‖Z

λn+1,µn+1;1

t

≤∫ t

0

∥∥Σn+1τ S0

−(t−τ)

∥∥Z

λn+1,µn+1;1

t

dτ

≤∫ t

0

∥∥Σn+1τ ‖

Zλn+1,µn+1;1τ

dτ,

and thus

d+

dt‖hn+1‖

Zλn+1,µn+1;1

t

≤ ‖Σn+1t ‖

Zλn+1,µn+1;1

t

.

Finally, according to Lemma 9.1, to this estimate we should add a negative multipleof the norm of ∇hn+1 to take into account the time-dependence of λn+1, µn+1.

All in all, after application of Proposition 4.24, we get

d+

dt

∥∥hn+1(t, · )∥∥Z

λn+1(t),µn+1(t);1

t

≤∥∥F [hn+1

t ]‖Fλn+1t+µn+1 ‖∇vfnt ‖Zλn+1,µn+1;1

t

+∥∥F [fn

t ]‖Fλn+1t+µn+1 ‖∇vhn+1t ‖

Zλn+1,µn+1;1

t

+∥∥F [hn

t ]‖Fλn+1t+µn+1 ‖∇vhnt ‖Zλn+1,µn+1;1

t

−K∥∥∇xh

n+1t ‖

Zλn+1,µn+1;1

t

−K ‖∇vhn+1t ‖

Zλn+1,µn+1;1

t

,

where K > 0, t is sufficiently small, and all exponents λn+1 and µn+1 in the right-hand side actually depend on t.

From Proposition 4.15 (iv) we easily get ‖F [h]‖Fλt+µ ≤ C ‖∇h‖Zλ,µ;1t

. Moreover,

by Proposition 4.10,

‖∇fn‖Z

λn+1,µn+1;1

t

≤∑

k≤n

‖∇hk‖Z

λn+1,µn+1;1

t

≤ C∑

k≤n

‖hk‖Z

λk+1,µk+1;1

t

minλk − λn+1 ; µk − µn+1

.


We end up with the bound

d+

dt

∥∥hn+1(t, · )∥∥Z

λn+1(t),µn+1(t);1

t

≤[C

(∑

k≤n

δk


)

−K

]∥∥∇hn+1

∥∥Z

λn+1(t),µn+1(t);1

t

+δ2n

minλn − λn+1 ; µn − µn+1

.

We conclude that if

(9.2)∑

k≤n

δk


≤ K

C,

then we may choose

(9.3) δn+1 =δ2n

minλn − λn+1 ; µn − µn+1

.

This is our first encounter with the principle of “stratification” of errors, whichwill be crucial in the next section: to control the error at stage n+1, we use not onlythe smallness of the error from stage n, but also an information about all previouserrors; namely the fact that the convergence of the size of the error is much fasterthan the convergence of the regularity loss. Let us see how this works. We chooseλk − λk+1 = µk − µk+1 = Λ/k2, where Λ > 0 is arbitrarily small. Then for k ≤ n,λk − λn+1 ≥ Λ/k2, and therefore δn+1 ≤ δ2

n n2/Λ. The problem is to check

(9.4)

∞∑

n=1

n2 δn < +∞.

Indeed, then we can choose K large enough for (9.2) to be satisfied, and then Tsmall enough that, say λ∗ − 2KT ≥ λ♯, µ∗ −KT ≥ µ♯, where λ♯ < λ∗, µ♯ < µ∗ havebeen fixed in advance.

If δ1 = δ, the general term in the series of (9.4) is

n2 δ2n

Λn(22)2n−1

(32)2n−2

(42)2n−2

. . . ((n− 1)2)2 n2.

To prove the convergence for δ small enough, we assume by induction that δn ≤ zan,

where a is fixed in the interval (1, 2) (say a = 1.5); and we claim that this condition


propagates if z > 0 is small enough. Indeed,

δn+1 ≤z2 an

Λn2 ≤ zan+1

(z(2−a)an

n2

Λ

),

and this is bounded above by zan+1if z is so small that

∀n ∈ N, z(2−a) an ≤ Λ

n2.

This concludes the iteration argument. Note that the convergence is still extremelyfast — like O(zan

) for any a < 2. (Of course, when a approaches 2, the constantsbecome huge, and the restriction on the size of the perturbation becomes more andmore stringent.)

Remark 9.3. The method used in this section can certainly be applied to moregeneral situations of Cauchy–Kowalevskaya type. Actually, as pointed out to usby Bony and Gerard, the use of a regularity index which decays linearly in time,combined with a Newton iteration, was used by Nirenberg [71] to prove an abstractCauchy–Kowalevskaya theorem. Nirenberg uses a time-integral formulation, so thereis nothing in [71] comparable to Lemma 9.1, and the details of the proof of conver-gence differ from ours; but the general strategy is similar. Nirenberg’s proof waslater simplified by Nishida [72] with a clever fixed point argument; in the presentsection anyway, our final goal is to provide short-term estimates for the successivecorrections arising from the Newton scheme.

10. Global in time iteration

Now let us implement the scheme described in Section 8, with some technicalmodifications. If f is a given kinetic distribution, we write ρ[f ] =

∫f dv and F [f ] =

−∇W ∗ ρ[f ]. We let

(10.1) fn = f 0 + h1 + . . .+ hn,

where the successive corrections hk are defined by the natural Newton scheme intro-duced in Section 8. As in Section 5 we define Ωk

t,τ as the deflection from time t to

time τ , generated by the force field F [fk] = −∇W ∗ ρ[fk]. (Note that Ω0 = Id .)

10.1. The statement of the induction. We shall fix p ∈ [1,∞] and make thefollowing assumptions:• Regularity of the background: there are λ > 0 and C0 > 0 such that

∀p ∈ [1, p], ‖f 0‖Cλ;p ≤ C0.


• Linear damping condition: The stability condition (L) from Subsection 2.2 holdswith parameters C0, λ, (the same as above) and κ > 0.

• Regularity of the interaction: There are γ > 1 and CF > 0 such that for anyν > 0,

(10.2)∥∥∇W ∗ ρ

∥∥Fν,γ ≤ CF ‖ρ‖Fν .

• Initial layer of regularity (coming from Section 9): Having chosen λ♯ < λ, µ♯ < µ,we assume that for all p ∈ [1, p]

(10.3) ∀ k ≥ 1, sup0≤t≤T

(‖hk

t ‖Zλ♯,µ♯;p + ‖ρ[hkt ]‖Fµ♯

)≤ ζk,

where T is some positive time, and ζk converges to zero extremely fast: ζk = O(zak

II ),

zI ≤ C δ < 1, 1 < aI < 2 (aI chosen in advance, arbitrarily close to 2).

• Smallness of the solution of the linearized equation (coming from Subsection4.12): Given λ1 < λ♯, µ1 < µ♯, we assume

(10.4) ∀ p ∈ [1, p],

supτ≥0

∥∥ρ[h1τ ]∥∥Fλ1τ+µ1

≤ δ1

supt≥τ≥0

∥∥h1τ

∥∥Z

λ1(1+b),µ1;p

τ− bt1+b

≤ δ1,

where δ1 ≤ C δ.

Then we prove the following induction: for any n ≥ 1,

(10.5) ∀ k ∈ 1, . . . , n, ∀ p ∈ [1, p],

supτ≥0

∥∥ρ[hkτ ]∥∥Fλkτ+µk

≤ δk

supt≥τ≥0

∥∥∥hkτ Ωk−1

t,τ

∥∥∥Z

λk(1+b),µk;p

τ− bt1+b

≤ δk,

where

• (δk)k∈N is a sequence satisfying 0 < CF ζk ≤ δk, and δk = O(zak), z < zI ,

1 < a < aI (a arbitrarily close to aI),

• (λk, µk) are decreasing to (λ∞, µ∞), where (λ∞, µ∞) are arbitrarily close to(λ1, µ1); in particular we impose

(10.6) λ♯ − λ∞ ≤ min

1 ;λ∞2

, µ♯ − µ∞ ≤ min

1 ;

µ∞

2

.


• T is some small positive time in (10.3); we impose

(10.7) λ# T ≤ µ♯ − µ1

2.

• b = b(t) =B

1 + t, where B ∈ (0, T ) is a (small) constant.

10.2. Preparatory remarks. As announced in (10.5), we shall propagate the fol-lowing “primary” controls on the density and distribution:

(10.8) (En

ρ ) ∀ k ∈ 1, . . . , n, supτ≥0

∥∥ρ[hkτ ]∥∥Fλkτ+µk

≤ δk

and

(10.9) (En

h) ∀ k ∈ 1, . . . , n, ∀ p ∈ [1, p], sup

t≥τ≥0

∥∥∥hkτ Ωk−1

t,τ

∥∥∥Z

λk(1+b),µk ;p

τ− bt1+b

≤ δk.

Estimate (En

ρ ) obviously implies, via (10.2), up to a multiplicative constant,

(10.10) (En

ρ ) ∀ k ∈ 1, . . . , n, supτ≥0

∥∥F [hkτ ]∥∥Fλkτ+µk,γ ≤ δk.

Before we can go from there to stage n+1, we need an additional set of estimateson the deflection maps (Ωk)k=1,...,n, which will be used to

(1) update the control on Ωkt,τ − Id ;

(2) establish the needed control along the characteristics for the background(∇vf

nτ ) Ωn

t,τ (same index for the distribution and the deflection);

(3) update some technical controls allowing to exchange (asymptotically) gradi-ent and composition by Ωk

t,τ ; this will be crucial to handle the contributionof the zero mode of the background after composition by characteristics.

This set of deflection estimates falls into three categories. The first group expressesthe closeness of Ωk to Id :(10.11)

(En

Ω) ∀ k ∈ 1, . . . , n,

supt≥τ≥0

∥∥∥ΩkXt,τ − Id∥∥∥Z

λ∗k(1+b),(µ∗

k,γ)

τ− bt1+b

≤ 2Rk2(τ, t),

supt≥τ≥0

∥∥∥ΩkVt,τ − Id∥∥∥Z

λ∗k(1+b),(µ∗

k,γ)

τ− bt1+b

≤ Rk1(τ, t),


with λk > λ∗k > λk+1, µk > µ∗k > µk+1, and

(10.12)

Rk1(τ, t) =

(k∑

j=1

δj e−2π(λj−λ∗

j )τ

2π(λj − λ∗j )

)min (t− τ) ; 1

Rk2(τ, t) =

(k∑

j=1


j )τ

(2π(λj − λ∗j))2

)min

(t− τ)2

2; 1

.

The second group of estimates expresses the fact that Ωn −Ωk is very small whenk is large:(10.13)

(En

Ω) ∀ k ∈ 0, . . . , n−1,

supt≥τ≥0

∥∥∥ΩnXt,τ − ΩkXt,τ

∥∥∥Z

λ∗n(1+b),(µ∗

n,γ)

τ− bt1+b

≤ 2Rk,n2 (τ, t),

supt≥τ≥0

∥∥∥ΩnVt,τ − ΩkVt,τ

∥∥∥Z

λ∗n(1+b),(µ∗

n,γ)

τ− bt1+b

≤ Rk,n1 (τ, t) + Rk,n

2 (τ, t),

supt≥τ≥0

∥∥∥(Ωkt,τ )

−1 Ωnt,τ − Id

∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ 4(Rk,n

1 (τ, t) + Rk,n2 (τ, t)

),

with

(10.14)

Rk,n1 (τ, t) =

(n∑

j=k+1


j )τ

2π(λj − λ∗j )

)min (t− τ) ; 1

Rk,n2 (τ, t) =

(n∑

j=k+1


j )τ

(2π(λj − λ∗j))2

)min

(t− τ)2

2; 1

.

(Choosing k = 0 brings us back to the previous estimates (En

Ω).)

The last group of estimates expresses the fact that the differential of the deflectionmap is uniformly close to the identity (in a way which is more precise than whatwould follow from the first group of estimates):(10.15)

(En

∇Ω) ∀ k = 1, . . . , n,

supt≥τ≥0

∥∥∥∇ΩkXt,τ − (I, 0)∥∥∥Z

λ∗k(1+b),µ∗

k

τ− bt1+b

≤ 2Rk2(τ, t),

supt≥τ≥0

∥∥∥∇ΩkVt,τ − (0, I)∥∥∥Z

λ∗k(1+b),µ∗

k

τ− bt1+b

≤ Rk1(τ, t) + Rk

2(τ, t),

where ∇ = (∇x,∇v), and I is the identity matrix.

An important property of the functions Rk,n1 (τ, t), Rk,n

2 (τ, t) is their fast decay asτ → ∞ and as k → ∞, uniformly in n ≥ k; this is due to the fast convergence of


the sequence (δk)k∈N. Eventually, if r ∈ N is given, we shall have

(10.16) ∀ r ≥ 1, Rk,n1 (τ, t) ≤ ωr,1

k,n(τ, t), Rk,n2 (τ, t) ≤ ωr,2

k,n(τ, t)

with

ωr,1k,n(τ, t) := Cr

ω

(n∑

j=k+1

δj(2π(λj − λ∗j ))

1+r

)min (t− τ) ; 1

(1 + τ)r,

and

ωr,2k,n(τ, t) := Cr

ω

(n∑

j=k+1


2+r

)min (t− τ)2/2 ; 1

(1 + τ)r

for some absolute constant Crω depending only on r (we also denote ωr,1

0,n = ωr,1n and

ωr,20,n = ωr,2

n ).

From the estimates on the characteristics and (En

h) will follow the following “sec-

ondary controls” on the distribution function:

(10.17) (En

h) ∀ k ∈ 1, . . . , n, ∀ p ∈ [1, p],

supt≥τ≥0

∥∥∥(∇xhkτ ) Ωk−1

t,τ

∥∥∥Z

λk(1+b),µk ;p

τ− bt1+b

≤ δk

supt≥τ≥0

∥∥∥∇x

(hk

τ Ωk−1t,τ

)∥∥∥Z

λk(1+b),µk ;p

τ− bt1+b

≤ δk

∥∥∥((∇v + τ∇x

)hk

τ ) Ωk−1t,τ

∥∥∥Z

λk(1+b),µk ;p

τ− bt1+b

≤ δk

∥∥∥(∇v + τ∇x)(hk

τ Ωk−1t,τ

)∥∥∥Z

λk(1+b),µk ;p

τ− bt1+b

≤ δk

supt≥τ≥0

1

(1 + τ)2

∥∥∥(∇∇hk

τ

) Ωk−1

t,τ

∥∥∥Z

λk(1+b),µk ;1

τ− bt1+b

≤ δk

supt≥τ≥0

(1 + τ)2∥∥∥(∇hk

τ ) Ωk−1t,τ −∇

(hk

τ Ωk−1t,τ

)∥∥∥Z

λk(1+b),µk ;1

τ− bt1+b

≤ δk.

The transition from stage n to stage n+ 1 can be summarized as follows:

(En

ρ )(An)=⇒

[(En

Ω) + (En

Ω) + (En

∇Ω)]

[(En

ρ ) + (En

Ω) + (En

Ω) + (En

∇Ω) + (En

h) + (En

h)]

(Bn)=⇒

[(En+1

ρ ) + (En+1

ρ ) + (En+1

h) + (En+1

h)].


The first implication (An) is proven by an amplification of the technique usedin Section 5; ultimately, it relies on repeated application of Picard’s fixed pointtheorem in analytic norms. The second implication (Bn) is the harder part; ituses the machinery from Sections 6 and 7, together with the idea of propagatingsimultaneously a shifted Z norm for the kinetic distribution and an F norm for thedensity.

In both implications, the stratification of error estimates will prevent the blow upof constants. So we shall decompose the force field F n generated by fn as

F n = F [fn] = E1 + . . .+ En,

where Ek = F [hk] = −∇W ∗ ρ[hk].The plan of the estimates is as follows. We shall construct inductively a sequence

of constant coefficients

λ♯ > λ1 > λ∗1 > λ2 > . . . > λn > λ∗n > λn+1 > . . .

µ♯ > µ1 > µ∗1 > µ2 > . . . > µn > µ∗

n > µn+1 > . . .

(where λn, µn will be fixed in the proof of (An), and λn+1, µn+1 in the proof of (Bn))converging respectively to λ∞ and µ∞; and a sequence (δk)k∈N decreasing very fastto zero. For simplicity we shall let

Rn(τ, t) = Rn1 (τ, t) + Rn

2 (τ, t), Rk,n(τ, t) = Rk,n1 (τ, t) + Rk,n

2 (τ, t),

and assume 2π(λj − λ∗j ) ≤ 1; so(10.18)

Rk,n(τ, t) ≤ Crω

(n∑

j=k+1

δj(2π(λj − λ∗j))

2+r

)mint− τ ; 1

(1 + τ)r, R0,n = Rn.

It will be sufficient to work with some fixed r, large enough (as we shall see, r = 4will do).

To go from stage n to stage n+ 1, we shall do as follows:

- Implication (An) (subsection 10.3):

Step 1. estimate Ωn − Id (the bound should be uniform in n);

Step 2. estimate Ωn − Ωk (k ≤ n− 1; the error should be small when k → ∞);

Step 3. estimate ∇Ωn − I;

Step 4. estimate (Ωk)−1 Ωn;

- Implication (Bn) (subsection 10.4):

Step 5. estimate hk and its derivatives along the composition by Ωn;


Step 6. estimate ρ[hn+1], using Sections 6 and 7;

Step 7. estimate F [hn+1] from ρ[hn+1];

Step 8. estimate hn+1 Ωn;

Step 9. estimate derivatives of hn+1 composed with Ωn;

Step 10. show that for hn+1, ∇ and composition by Ωn asymptotically commute.

10.3. Estimates on the characteristics. In this subsection, we assume that esti-

mate (En

ρ ) is proven, and we establish (En

Ω) + (En

Ω) + (En

∇Ω). Let λ∗n < λn, µ

∗n < µn

to be fixed later on.

10.3.1. Step 1: Estimate of Ωn − Id . This is the first and archetypal estimate. We

shall bound ΩnXt,τ − x in the hybrid norm Zλ∗n(1+b),µ∗

n

τ− bt1+b

. The Sobolev correction

γ will play no role here in the proofs, and for simplicity we shall forget it in thecomputations, just recall it in the final results. (Use Proposition 4.32 wheneverneeded.)

Since we expect the characteristics for the force field F n to be close to the freetransport characteristics, it is natural to write

(10.19) Xnt,τ (x, v) = x− v(t− τ) + Zn

t,τ (x, v),

where Znt,τ solves

(10.20)

∂2

∂τ 2Zn

t,τ (x, v) = F n(τ, x− v(t− τ) + Zn

t,τ (x, v))

Znt,t(x, v) = 0, ∂τZ

nt,τ

∣∣∣t=τ

(x, v) = 0.

(With respect to Section 5 we have dropped the parameter ε, to take advantage ofthe “stratified” nature of F n; anyway this parameter was cosmetic.) So if we fixt > 0, (Zn

t,τ ) is a fixed point of the map

Ψ : (Wt,τ )0≤τ≤t 7−→ (Zt,τ )0≤τ≤t

defined by

(10.21)

∂2

∂τ 2Zt,τ = F n

(τ, x− v(t− τ) +Wt,τ

)

Zt,t = 0, ∂τZt,τ

∣∣∣τ=t

= 0.

The goal is to estimate Znt,τ − x in the hybrid norm Zλ∗

n(1+b),µ∗n

t− bt1+b

.


We first bound (Zn0 )t,τ = Ψ(0). Explicitly,

(Zn0 )t,τ (x, v) =

∫ t

τ

(s− τ)F n(s, x− v(t− s)

)ds.

By Propositions 4.15 (i) and 4.19,

∥∥(Zn0 )t,τ

∥∥Z

λ∗n(1+b),µ∗

n

t− bt1+b

(10.22)

≤∫ t

τ

(s− τ)∥∥∥F n

(s, x− v(t− s)

)∥∥∥Z

λ∗n(1+b),µ∗

n

t− bt1+b

=

∫ t

τ

(s− τ) ‖F n(s, · )‖Z

λ∗n(1+b),µ∗

n

s− bt1+b

=

∫ t

τ

(s− τ) ‖F n(s, · )‖Fν(s,t) ds,

where

ν(s, t) = λ∗n∣∣s− b(t− s)

∣∣+ µ∗n.(10.23)

First case: If s ≥ bt/(1 + b), then

(10.24) ν(s, t) ≤ λ∗ns+ µ∗n ≤ λk s+ µk − (λk − λ∗n)s (1 ≤ k ≤ n).

Second case: If s < bt/(1+ b), then necessarily s ≤ B ≤ T . Taking into account(10.6), we have

ν(s, t) = λ∗n bt+ µ∗n − λ∗n(1 + b)s(10.25)

≤ λ∗nB + µ∗n − (λk − λ∗n)s.(10.26)

(Of course, the assumption λ♯ − λ∞ ≤ min1, λ∞/2 implies λk − λ∗n ≤ λ∗n.) Inparticular, by (10.7),

(10.27) ν(s, t) ≤ µ♯ − (λk − λ∗n)s (1 ≤ k ≤ n).


We plug these bounds into (10.22), then use Ek(s, 0) = 0 and the bounds (10.10)and (10.24) (for large times), and (10.3) and (10.27) (for short times). This yields

‖(Zn0 )t,τ‖Zλ∗

n(1+b),µ∗n

t− bt1+b

(10.28)

≤n∑

k=1

(∫ t

τ∨ bt1+b

(s− τ) ‖Ek(s, · )‖Fλks+µk−(λk−λ∗n)s ds

+

∫ τ∨ bt1+b

τ

(s− τ) ‖Ek(s, · )‖Fµ♯−(λk−λ∗

n)s ds

)

≤n∑

k=1

(∫ t

τ∨ bt1+b

(s− τ) e−2π(λk−λ∗n)s ‖Ek(s, · )‖Fλks+µk ds

+

∫ τ∨ bt1+b

τ

(s− τ) e−2π(λk−λ∗n)s ‖Ek(s, · )‖

Fµ♯ ds

)

≤n∑

k=1

δk

∫ t

τ

(s− τ) e−2π(λk−λ∗n)s ds

≤n∑

k=1

δk e−2π(λk−λ∗

n)τ min

(t− τ)2

2;

1

(2π(λk − λ∗n))2

≤ Rn

2 (τ, t).

Let us define the norm

∥∥∥∥∥∥∥∥(Zt,τ )0≤τ≤t

∥∥∥∥∥∥∥∥

n

:= sup0≤τ≤t

‖Zt,τ‖Zλ∗n(1+b),µ∗

n

t− bt1+b

Rn2 (τ, t)

.

(Note the difference with Section 5: now the regularity exponents depend on time(s).)Inequality (10.28) means that ‖‖Ψ(0)‖‖n ≤ 1. We shall check that Ψ is (1/2)-Lipschitz on the ball B(0, 2) in the norm ‖‖ · ‖‖n. This will be subtle: the uniformbounds on the size of the force field, coming from the preceding steps, will allow toget good decaying exponentials, which in turn will imply uniform error bounds atthe present stage.


So let W, W ∈ B(0, 2), and let Z = Ψ(W ), Z = Ψ(W ). As in Section 5, we write

Zt,τ − Zt,τ =

∫ 1

0

∫ t

τ

(s− τ)∇xFn(s, x− v(t− s) +

(θWt,s + (1 − θ) Wt,s

))

· (Wt,s − Wt,s) ds dθ,

and deduce ∥∥∥∥∥∥∥∥(Zt,τ − Zt,τ

)0≤τ≤t

∥∥∥∥∥∥∥∥

n

≤ A(t)

∥∥∥∥∥∥∥∥(Wt,s − Wt,s

)0≤s≤t

∥∥∥∥∥∥∥∥

n

,

where

A(t) = sup0≤τ≤s≤t

Rn2 (s, t)

Rn2 (τ, t)

×∫ 1

0

∫ t

τ

(s− τ)∥∥∥∇xF

n(s, x− v(t− s)+

(θWt,s + (1 − θ) Wt,s

))∥∥∥Z

λ∗n(1+b),µ∗

n

t− bt1+b

ds dθ.

For τ ≤ s we have Rn2 (s, t) ≤ Rn

2 (τ, t). Also, by Propositions 4.25 (applied withV = 0, b = −(t− s) and σ = 0 in that statement) and 4.15,

A(t) ≤ sup0≤τ≤t

∫ t

τ

(s− τ) ‖∇xFn(s, · )‖Fν(s,t)+e(s,t) ds,

where ν is defined by (10.23) and the “error” e(s, t) arising from composition is givenby

e(s, t) = sup0≤θ≤1

∥∥∥θWt,s + (1 − θ) Wt,s

∥∥∥Z

λ∗n(1+b),µ∗

n

t− bt1+b

≤ 2Rn2 (s, t).

Since

Rn2 (s, t) ≤ ω1,2

n (s, t) := C1ω

(n∑

k=1

δk(2π(λk − λ∗k))

3

)min (t− s)2/2 ; 1

(1 + s)

we have, for all 0 ≤ s ≤ t,

(10.29) 2Rn2 (s, t) ≤ λ∗n

2b (t− s) 1s≥bt/(1+b) +

µ♯ − µ∗n

21s≤bt/(1+b),

as soon as(10.30)

(C1) ∀n ≥ 1, 2C1ω

(n∑

k=1

δk(2π(λk − λ∗n))3

)≤ min

λ∗nB

6;µ♯ − µ∗

n

2

.


We shall check later in Subsection 10.5 the feasibility of condition (C1) — as wellas a number of other forthcoming ones.

The extra error term in the exponent is sufficiently small to be absorbed by whatwe throw away in (10.24) or (10.25)-(10.26)-(10.27). So we obtain, as in the estimateof Zn

0 , for any k ∈ 1, . . . , n,

(ν + e)(s, t)

≤ λk s+ µk − (λk − λ∗n) s for s ≥ bt/(1 + b)

≤ µ♯ − (λk − λ∗n) s for s ≤ bt/(1 + b),

and we deduce (using (10.10) and γ ≥ 1)

A(t) ≤ sup0≤τ≤t

n∑

k=1

(∫ t

τ∨ bt1+b

(s− τ) ‖∇xEk(s, · )‖Fλks+µk−(λk−λ∗

n) s ds

+

∫ τ∨ bt1+b

τ

(s− τ) ‖∇xEk(s, · )‖

Fµ♯−(λk−λ∗n)s ds

)

≤ sup0≤τ≤t

n∑

k=1

δk

∫ t

τ

(s− τ) e−(λk−λ∗n) s ds ≤ sup

0≤τ≤tRn

2 (τ, t) = Rn2 (0, t)

≤n∑

k=1


.

If the latter quantity is bounded above by 1/2, then Ψ is (1/2)-Lipschitz and wemay apply the fixed point result from Theorem A.2. Therefore, under the condition(whose feasibility will be checked later)

(10.31) (C2) ∀n ≥ 1,n∑

k=1


≤ 1

2

we deduce

‖Znt,τ‖Zλ∗

n(1+b),µ∗n

t− bt1+b

≤ 2Rn2(τ, t).

After that, the estimates on the deflection map are obtained exactly as in Section5: writing Ωn

t,τ = (ΩnXt,τ ,ΩnVt,τ ), recalling the dependence on γ again, we end up


with

(10.32)

∥∥∥ΩnXt,τ − x∥∥∥Z

λ∗n(1+b),(µ∗

n,γ)

τ− bt1+b

≤ 2Rn2 (τ, t)

∥∥∥ΩnVt,τ − v∥∥∥Z

λ∗n(1+b),(µ∗

n,γ)

τ− bt1+b

≤ Rn1 (τ, t).

10.3.2. Step 2: Estimate of Ωn − Ωk. In this step our goal is to estimate Ωn − Ωk

for 1 ≤ k ≤ n− 1. The point is that the error should be small as k → ∞, uniformlyin n, so we can’t just write ‖Ωn − Ωk‖ ≤ ‖Ωn − Id ‖ + ‖Ωk − Id ‖. Instead, we startagain from the differential equation satisfied by Zk and Zn:

∂2

∂τ 2

(Zn

t,τ − Zkt,τ

)(x, v)

= F n(τ, x− v(t− τ) + Zn

t,τ (x, v))− F k

(τ, x− v(t− τ) + Zk

t,τ (x, v))

=

[F n(τ, x− v(t− τ) + Zn

t,τ

)− F n

(τ, x− v(t− τ) + Zk

t,τ

)]

+ (F n − F k)(τ, x− v(t− τ) + Zk

t,τ

).

This, together with the boundary conditions Znt,t − Zk

t,t = 0, ∂τ (Znt,τ − Zk

t,τ )|τ=t = 0,implies

Znt,τ − Zk

t,τ

=

∫ 1

0

∫ t

τ

(s− τ)∇xFn(s, x− v(t− s) +

(θ Zk

t,s + (1 − θ)Znt,s

))· (Zn

t,s − Zkt,s) ds dθ

+

∫ t

τ

(s− τ) (F n − F k)(s, x− v(t− s) + Zk

t,s(x, v))ds.

We fix t and define the norm

∥∥∥∥∥∥∥∥(Zt,τ )0≤τ≤t

∥∥∥∥∥∥∥∥

k,n

:= sup0≤τ≤t

‖Zt,τ‖Zλ∗n(1+b),µ∗

n

t− bt1+b

Rk,n2 (τ, t)

,


where Rk,n2 is defined in (10.14). Using the bounds on Zn, Zk in

∥∥∥∥ ·∥∥∥∥

n(since∥∥∥∥ ·

∥∥∥∥n≤∥∥∥∥ ·∥∥∥∥

kby using the fact that Rk

2 ≤ Rn2 ) and proceeding as before, we get

(10.33)∥∥∥∥∥∥(Zn

t,τ − Zkt,τ

)0≤τ≤t

∥∥∥∥∥∥

k,n≤ 1

2

∥∥∥∥∥∥∥∥(Zn

t,τ − Zkt,τ

)0≤τ≤t

∥∥∥∥∥∥∥∥

k,n

+∥∥∥∥∥∥(∫ t

τ

(s− τ) (F n − F k)(s, x− v(t− s) + Zk

t,s

)ds

)

0≤τ≤t

∥∥∥∥∥∥

k,n.

Next we estimate∥∥∥(F n − F k)(s, x− v(t− s) + Zk

t,s

)∥∥∥Z

λ∗n(1+b),µ∗

n

t− bt1+b

=∥∥∥(F n − F k)(s,Xk

t,s)∥∥∥Z

λ∗n(1+b),µ∗

n

t− bt1+b

=∥∥∥(F n − F k)(s,Ωk

t,s)∥∥∥Z

λ∗n(1+b),µ∗

n

s− bt1+b

≤∥∥∥(F n − F k)(s, · )

∥∥∥Fν(s,t)+e(s,t)

,

where the last inequality follows from Proposition 4.25, ν is again given by (10.23),and

e(s, t) =∥∥ΩkXt,s − Id

∥∥Z

λ∗n(1+b),µ∗

n

s− bt1+b

≤ 2Rk2(s, t) ≤ 2Rn

2 (s, t).

The same reasoning as in Step 1 yields, under assumptions (C1)-(C2), for k+1 ≤j ≤ n:

(ν + e)(s, t)

≤ λj s+ µj − (λj − λ∗n) s for s ≥ bt/(1 + b)

≤ µ♯ − (λj − λ∗n) s for s ≤ bt/(1 + b),

and so∥∥F n

s − F ks

∥∥Fν+e ≤

n∑

j=k+1


n) s.

For any τ ≥ 0, by integrating in time we find∥∥∥∫ t

τ

(s− τ) (F n − F k)(s, x− v(t− s) + Zk

t,s

)ds∥∥∥Z

λ∗n(1+b),µ∗

n

t− bt1+b

≤∫ t

τ

(s− τ)n∑

j=k+1


n) s ds ≤ Rk,n2 (τ, t).


Therefore∥∥∥∥∥∥∥∥(∫ t

τ

(s− τ) (F n − F k)(s, x− v(t− s) + Zk

t,s

)ds

)

0≤τ≤t

∥∥∥∥∥∥∥∥

k,n

≤ 1

and by (10.33) ∥∥∥∥∥∥∥∥(Zn

t,τ − Zkt,τ

)0≤τ≤t

∥∥∥∥∥∥∥∥

k,n

≤ 2.

Recalling the Sobolev correction, we conclude that

(10.34)∥∥ΩnXt,τ − ΩkXt,τ

∥∥Z

λ∗n(1+b),(µ∗

n,γ)

τ− bt1+b

≤ 2Rk,n2 (τ, t).

For the velocity component, say U , we write

∂

∂τ

(Un

t,τ − Ukt,τ

)(x, v)

= F n(τ, x− v(t− τ) + Zn

t,τ (x, v))− F k

(τ, x− v(t− τ) + Zk

t,τ (x, v))

=

[F n(τ, x− v(t− τ) + Zn

t,τ

)− F n

(τ, x− v(t− τ) + Zk

t,τ

)]

+ (F n − F k)(τ, x− v(t− τ) + Zk

t,τ

).

where Zn, Zk were estimated above, and the boundary conditions are Unt,t −Uk

t,t = 0.Thus

Unt,τ − Uk

t,τ

=

∫ 1

0

∫ t

τ

∇xFn(s, x− v(t− s) +

(θ Zk

t,s + (1 − θ)Znt,s

))· (Zn

t,s − Zkt,s) ds dθ

+

∫ t

τ

(F n − F k)(s, x− v(t− s) + Zk

t,s(x, v))ds,

and from this one easily derives the similar estimates

∥∥ΩnXt,τ − ΩkXt,τ

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ 2Rk,n2 (t, τ)

∥∥ΩnVt,τ − ΩkVt,τ

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ Rk,n1 (t, τ) + Rk,n

2 (t, τ).


10.3.3. Step 3: Estimate of ∇Ωn. Now we establish a control on the derivative ofthe deflection map. Of course, we could deduce such a control from the bound onΩn − Id and Proposition 4.32(vi): for instance, if λ∗∗n < λ∗n, µ

∗∗n < µ∗

n, then

(10.35)∥∥∇Ωn

t,τ − I∥∥Z

λ∗∗n (1+b),(µ∗∗

n ,γ)

τ− bt1+b

≤ CRn2 (τ, t)

minλ∗n − λ∗∗n ; µ∗

n − µ∗∗n

.

But this bound involves very large constants, and is useless in our argument.Better estimates can be obtained by using again the equation (10.20). Writing

(Ωnt,τ − Id )(x, v) =

(Zn

t,τ

(x+ v(t− τ), v

), Zn

t,τ

(x+ v(t− τ), v

)),

where the dot stands for ∂/∂τ , we get by differentiation

∇xΩnt,τ − (I, 0) =

(∇xZ

nt,τ

(x+ v(t− τ), v

), ∇xZ

nt,τ

(x+ v(t− τ), v

)),

∇vΩnt,τ−(0, I) =

((∇v+(t−τ)∇x)Z

nt,τ

(x+v(t−τ), v

), (∇v+(t−τ)∇x)Z

nt,τ

(x+v(t−τ), v

)).

Let us estimate for instance ∇xΩ−(I, 0), or equivalently ∇xZnt,τ . By differentiating

(10.20), we obtain

∂2

∂τ 2∇xZ

nt,τ (x, v) = ∇xF

n(τ, x− v(t− τ) + Zn

t,τ (x, v))· (Id + ∇xZ

nt,τ ).

So ∇xZnt,τ is a fixed point of Ψ : W 7−→ Q, where W and Q are functions of τ ∈ [0, t]

satisfying

∂2Q

∂τ 2= ∇xF

n(τ, x− v(t− τ) + Zn

t,τ

)(I +W ),

Q(t) = 0, ∂τQ(t) = 0.

We treat this in the same way as in Steps 1 and 2, and find on Qx (the x componentof Q) the same estimates as we had previously on the x component of Ω. Forthe velocity component, a direct estimate from the integral equation expressing thevelocity in terms of F yields a control by Rn

1 + Rn2 . Finally for ∇vΩ this is similar,

noting that (∇v + (t − τ)∇x)(x − v(t − τ)) = 0, the differential equation being forinstance:

∂2

∂τ 2(∇v+(t−τ)∇x)Z

nt,τ (x, v) = ∇xF

n(τ, x−v(t−τ)+Zn

t,τ (x, v))·((∇v+(t−τ)∇x)Z

nt,τ ).


In the end we obtain

(10.36)

supt≥τ≥0

∥∥∥∇ΩnXt,τ − (I, 0)∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ 2Rn2 (τ, t),

supt≥τ≥0

∥∥∥∇ΩnVt,τ − (0, I)∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ Rn1 (τ, t) + Rn

2 (τ, t).

10.3.4. Step 4: Estimate of (Ωk)−1 Ωn. We do this by applying Proposition 4.28with F = Ωk, G = Ωn. (Note: we cannot exchange the roles of Ωk and Ωn in thisstep, because we have a better information on the regularity of Ωk.) Let ε = ε(d)be the small constant appearing in Proposition 4.28. If

(10.37) (C3) ∀ k ≥ 1, 3Rk2(τ, t) + Rk

1(τ, t) ≤ ε,

then ‖∇Ωkt,τ − I‖

Zλ∗

k(1+b),µ∗

kτ−bt/(1+b)

≤ ε; if in addition

(10.38) (C4) ∀ k ∈ 1, . . . , n− 1, ∀ t ≥ τ,

2(1 + τ) (1 +B)(3Rk,n

2 + Rk,n1

)(τ, t) ≤ max

λ∗k − λ∗n ; µ∗

k − µ∗n

,

then

λ∗n(1 + b) + 2 ‖Ωn − Ωk‖Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ λ∗k(1 + b)

µ∗n + 2

(1 +

∣∣∣∣τ −bt

1 + b

∣∣∣∣)

‖Ωn − Ωk‖Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ µ∗k.

(Once again, short times should be treated separately. Further note that the need forthe factor (1+τ) in (C4) ultimately comes from the fact that we are composing alsoin the v variable, see the coefficient σ in the last norm of (4.30).) Then Proposition4.28 (ii) yields

∥∥∥(Ωkt,τ )

−1 Ωnt,τ − Id

∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ 2∥∥Ωk

t,τ − Ωnt,τ

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ 4 (Rk,n1 + Rk,n

2 )(τ, t).

10.3.5. Partial conclusion. At this point we have established (En

Ω) + (En

Ω) + (En

∇Ω).

10.4. Estimates on the density and distribution along characteristics. In

this subsection we establish (En+1

ρ ) + (En+1

ρ ) + (En+1

h) + (En+1

h).


10.4.1. Step 5: Estimate of hk Ωn and (∇hk) Ωn (k ≤ n). Let k ∈ 1, . . . , n.Since

hkτ Ωn

t,τ =(hk

τ Ωk−1t,τ

)((Ωk−1

t,τ )−1 Ωnt,τ

),

the control on hk Ωn will follow from the control on hk Ωk−1 in (En

h), together

with the control on (Ωk−1)−1 Ωn in (En

Ω). If

(10.39) (1 + τ)∥∥(Ωk−1

t,τ )−1 Ωnt,τ − Id

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ min(λk − λ∗n) ; (µk − µ∗

n),

then we can apply Proposition 4.25 and get, for any p ∈ [1, p], and t ≥ τ ≥ 0,

(10.40)∥∥∥hk

τ Ωnt,τ

∥∥∥Z

λ∗n(1+b),µ∗

n;p

τ− bt1+b

≤∥∥∥hk

τ Ωk−1t,τ

∥∥∥Z

λk(1+b),µk;p

τ− bt1+b

≤ δk.

In turn, (10.39) is satisfied if

(10.41) (C5) ∀ k ∈ 1, . . . , n, ∀ τ ∈ [0, t],

4 (1 + τ)(Rk,n

1 (τ, t) + Rk,n2 (τ, t)

)≤ min

λk − λ∗n ; µk − µ∗

n

;

we shall check later the feasibility of this condition.Then, by the same argument, we also have

∀ k ∈ 1, . . . , n, ∀ p ∈ [1, p],

supt≥τ≥0

∥∥∥(∇xhkτ ) Ωn

t,τ

∥∥∥Z

λ∗n(1+b),µ∗

n;p

τ− bt1+b

+∥∥∥((∇v + τ∇x)h

kτ

) Ωn

t,τ

∥∥∥Z

λ∗n(1+b),µ∗

n;p

τ− bt1+b

≤ δk.

10.4.2. Step 6: estimate on ρ[hn+1]. This step is the first where we shall use theVlasov equation. Starting from (8.4), we apply the method of characteristics to get,as in Section 8,

(10.42) hn+1(t, Xn

0,t(x, v), Vn0,t(x, v)

)=

∫ t

0

Σn+1(τ,Xn

0,τ(x, v), Vn0,τ (x, v)

)dτ,

where

Σn+1 = −(F [hn+1] · ∇vf

n + F [hn] · ∇vhn).

We compose this with (Xnt,0, V

nt,0) and apply (5.2) to get

hn+1(t, x, v) =

∫ t

0

Σn+1(τ,Xn

t,τ (x, v), Vnt,τ (x, v)

)dτ,


and so, by integration in the v variable,

ρ[hn+1](t, x) =

∫ t

0

∫

Rd

Σn+1(τ,Xn

t,τ (x, v), Vnt,τ(x, v)

)dv dτ(10.43)

= −∫ t

0

∫

Rd

(Rn+1τ,t ·Gn

τ,t)(x− v(t− τ), v) dv dτ

−∫ t

0

∫

Rd

(Rnτ,t ·Hn

τ,t)(x− v(t− τ), v) dv dτ,

where (with a slight inconsistency in the notation)

(10.44)

Rn+1

τ,t = F [hn+1] Ωnt,τ , Rn

τ,t = F [hn] Ωnt,τ ,

Gnτ,t = (∇vf

n) Ωnt,τ , Hn

τ,t = (∇vhn) Ωn

t,τ .

Since the free transport semigroup and Ωnt,τ are measure-preserving,

∀ 0 ≤ τ ≤ t,

∫

Td

∫

Rd

(Rn+1τ,t ·Gn

τ,t)(x− v(t− τ), v) dv dx

=

∫ ∫Rn+1

τ,t ·Gnτ,t dv dx

=

∫ ∫F [hn+1] · ∇vf

n dv dx

=

∫ ∫∇v ·

(F [hn+1] fn

)dv dx = 0,

and similarly

∀ 0 ≤ τ ≤ t,

∫

Td

∫

Rd

(Rnτ,t ·Hn

τ,t)(x− v(t− τ), v) dv dx = 0.

This will allow us to apply the inequalities from Section 6.

Substep a. Let us first deal with the source term

(10.45) σn,n(t, x) :=

∫ t

0

∫(Rn

τ,t ·Hnτ,t)(x− v(t− τ), v) dv dτ.

By Proposition 6.2,

(10.46)∥∥σn,n(t, · )

∥∥Fλ∗

nt+µ∗n≤∫ t

0

‖Rnτ,t‖Zλ∗

n(1+b),µ∗n

τ− bt1+b

‖Hnτ,t‖Zλ∗

n(1+b),µ∗n;1

τ− bt1+b

dτ.


On the one hand, we have from Step 5

‖Hnτ,t‖Zλ∗

n(1+b),µ∗n;1

τ− bt1+b

≤ 2 (1 + τ) δn.

On the other hand, under condition (C1), we may apply Proposition 4.25 (choos-ing σ = 0 in that proposition) to get

‖Rnτ,t‖Zλ∗

n(1+b),µ∗n

τ− bt1+b

≤∥∥F [hn

τ ]∥∥Fνn

,

where

νn(t, τ) = µ∗n + λ∗n(1 + b)

∣∣∣∣τ −bt

1 + b

∣∣∣∣+∥∥ΩnXt,τ − Id

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ µ∗n + λ∗n(1 + b)

∣∣∣∣τ −bt

1 + b

∣∣∣∣ + 2Rn2 (τ, t).

Proceeding as in Step 1 (treating small times separately), we deduce

‖Rnτ,t‖Zλ∗

n(1+b),µ∗n

τ− bt1+b

≤ ‖F [hnτ ]‖Fνn ≤ e−2π(λn−λ∗

n)τ ‖F [hnτ ]‖F νn

≤ CF e−2π(λn−λ∗

n)τ ‖ρ[hnτ ]‖F νn ≤ CF e

−2π(λn−λ∗n)τ δn,

with

(10.47)

νn(τ, t) := µ♯ when 0 ≤ τ ≤ bt/(1 + b)

νn(τ, t) := λnτ + µn when τ ≥ bt/(1 + b).

(We have used the gradient structure of the force to convert (gliding) regularity intodecay.) Thus

∫ t

0

‖Rnτ,t‖Zλ∗

n(1+b),µ∗n

τ− bt1+b

‖Hnτ,t‖Zλ∗

n(1+b),µ∗n;1

τ− bt1+b

dτ(10.48)

≤ 2CF δ2n

∫ t

0

e−2π(λn−λ∗n)τ (1 + τ) dτ

≤ 2CF δ2n

(π (λn − λ∗n))2.

(Note: This is the power 2 which is responsible for the very fast convergence of theNewton scheme.)


Substep b. Now let us handle the term

(10.49) σn,n+1(t, x) :=

∫ t

0

∫ (Rn+1

τ,t ·Gnτ,t

)(x− v(t− τ), v) dv dτ.

This is the focal point of all our analysis, because it is in this term that the self-consistent nature of the Vlasov equation appears. In particular, we will make crucialuse of the time-cheating trick to overcome the loss of regularity implied by composi-tion; and also the other bilinear estimates (regularity extortion) from Section 6, aswell as the time-response study from Section 7. Particular care should be given to thezero spatial mode of Gn, which is associated with instantaneous response (no echo).In the linearized equation we did not see this problem because the contribution ofthe zero mode was vanishing!

We start by introducing

(10.50) Gn

τ,t = ∇vf0 +

n∑

k=1

∇v

(hk

τ Ωk−1t,τ

),

and we decompose σn,n+1 as

(10.51) σn,n+1 = σn,n+1 + E + E ,

where

(10.52) σn,n+1(t, x) =

∫ t

0

∫F [hn+1

τ ] ·Gn

τ,t

(x− v(t− τ), v

)dv dτ

and the error terms E and E are defined by

(10.53) E(t, x) =

∫ t

0

∫ ((F [hn+1

τ ] Ωnt,τ − F [hn+1

τ ])·Gn

)(τ, x− v(t− τ), v

)dv dτ,

(10.54) E(t, x) =

∫ t

0

∫ (F [hn+1

τ ] ·(Gn −G

n))(τ, x− v(t− τ), v

)dv dτ.

We shall first estimate E and E .


Control of E : This is based on the time-cheating trick from Section 6, and theregularity of the force. By Proposition 6.2,

(10.55)∥∥E(t, · )

∥∥Fλ∗

nt+µ∗n≤∫ t

0

∥∥∥F [hn+1τ ] Ωn

t,τ − F [hn+1τ ]

∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

×

‖Gn‖Z

λ∗n(1+b),µ∗

n ;1

τ− bt1+b

dτ.

From (10.1) and Step 5,

‖Gn‖Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

≤∥∥∇vf

0 Ωnt,τ

∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

+

n∑

k=1

∥∥∇vhkτ Ωn

t,τ

∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

(10.56)

≤ C ′0 +

(n∑

k=1

δk

)(1 + τ),

where C ′0 comes from the contribution of f 0.

Next, by Propositions 4.24 and 4.25 (with V = 0, τ = σ, b = 0),

∥∥∥F [hn+1τ ] Ωn

t,τ − F [hn+1τ ]

∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

(10.57)

≤

∫ 1

0

∥∥∥∇F [hn+1τ ]

(Id + θ(Ωn

t,τ − Id ))∥∥∥

Zλ∗

n(1+b),µ∗n

τ− bt1+b

dθ

∥∥Ωn

t,τ − Id∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤∥∥∇F [hn+1

τ ]∥∥Fνn

∥∥Ωnt,τ − Id

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

,

where

νn = µ∗n + λ∗n(1 + b)

∣∣∣∣τ −bt

1 + b

∣∣∣∣ +∥∥ΩnXt,τ − x

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

.

Small times are taken care of, as usual, by the initial regularity layer, so we onlyfocus on the case τ ≥ bt/(1 + b); then

νn ≤(λ∗nτ + µ∗

n

)− λ∗n b(t− τ) + 2Rn(τ, t)

≤(λ∗nτ + µ∗

n

)− λ∗n

B (t− τ)

1 + t+ 4C1

ω

(n∑

k=1


3

)mint− τ ; 1

1 + τ.


To make sure that νn ≤ λ∗nτ + µ∗n, we assume that

(10.58) (C6) 4C1ω

n∑

k=1


3≤ λ∗∞B

3,

and we note that

mint− τ ; 11 + τ

≤ 3

(t− τ

1 + t

).

(This is easily seen by separating four cases: (a) t ≤ 2, (b) t ≥ 2 and t− τ ≤ 1, (c)t ≥ 2 and t− τ ≥ 1 and τ ≤ t/2, (d) t ≥ 2 and t− τ ≥ 1 and τ ≥ t/2.)

Then, since γ ≥ 1, we have∥∥∇F [hn+1

τ ]∥∥Fνn

≤∥∥∇F [hn+1

τ ]∥∥Fλ∗

nτ+µ∗n

(10.59)

≤∥∥F [hn+1

τ ]∥∥Fλ∗

nτ+µ∗n,γ

≤ CF

∥∥ρ[hn+1τ ]

∥∥Fλ∗

nτ+µ∗n.

(Note: Applying Proposition 4.10 instead of the regularity coming from the interac-tion would consume more regularity than we can afford to.)

Plugging this back into (10.57), we get∥∥∥F [hn+1


τ ]∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ 2Rn(τ, t)CF ‖ρ[hn+1τ ]‖Fλ∗

nτ+µ∗n

≤ 2C3ω CF

(n∑

k=1


5

)1

(1 + τ)3

∥∥ρ[hn+1τ ]

∥∥Fλ∗

nτ+µ∗n.

Recalling (10.53) and (10.56), applying Proposition 4.24, we conclude that

(10.60)∥∥E(t, · )

∥∥Fλ∗

nt+µ∗n≤ 2C3

ω CF

(C ′

0 +n∑

k=1

δk

)(n∑

k=1


5

)

∫ t

0

∥∥ρ[hn+1τ ]

∥∥Fλ∗

nτ+µ∗n

dτ

(1 + τ)2.

(We could be a bit more precise; anyway we cannot go further since we do not yethave an estimate on ρ[hn+1].)


b2. Control of E: This will use the control on the derivatives of hk. We start againfrom Proposition 6.2:

(10.61)∥∥E(t, · )

∥∥Fλ∗

nt+µ∗n≤∫ t

0

∥∥Gn −Gn∥∥

Zλ∗

n(1+b),µ∗n;1

τ− bt1+b

‖F [hn+1τ ]‖Fβn dτ,

where βn = λ∗n(1+ b)|τ − bt/(1+ b)|+µ∗n. We focus again on the case τ ≥ bt/(1+ b),

so that (with crude estimates)

‖F [hn+1τ ]‖Fβn ≤ ‖F [hn+1

τ ]‖Fλ∗nτ+µ∗

n ≤ CF ‖ρ[hn+1τ ]‖Fλ∗

nτ+µ∗n ,

and the problem is to control Gn −Gn:

(10.62)∥∥Gn −G

n∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

≤∥∥∥(∇vf

0) Ωnt,τ −∇vf

0∥∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

+n∑

k=1

∥∥∥(∇vhkτ )Ωn

t,τ−(∇vhkτ )Ωk−1

t,τ

∥∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

+n∑

k=1

∥∥∥(∇vhkτ )Ωk−1

t,τ −∇v

(hk

τΩk−1t,τ

)∥∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

.

By induction hypothesis (En

h), and since the Zλ,µ

τ norms are increasing as a func-tion of λ, µ,

n∑

k=1

∥∥∥(∇vhkτ ) Ωk−1

t,τ −∇v

(hk

τ Ωk−1t,τ

)∥∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

≤(

n∑

k=1

δk

)1

(1 + τ)2.

It remains to treat the first and second terms in the right-hand side of (10.62).This is done by inversion/composition as in Step 5; let us consider for instance thecontribution of hk, k ≥ 1:∥∥∥∇vh

kτ Ωn

t,τ −∇vhkτ Ωk−1

t,τ

∥∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

≤∫ 1

0

∥∥∥∇∇vhkτ ((1 − θ)Ωn

t,τ + θΩk−1t,τ

)∥∥∥Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b

∥∥∥Ωnt,τ − Ωk−1

t,τ

∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

dθ

≤ 2∥∥∥∇∇vh

kτ Ωk−1

t,τ

∥∥∥Z

λ∗k(1+b),µ∗

k;1

τ− bt1+b

∥∥∥Ωnt,τ − Ωk−1

t,τ

∥∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ 4 δk (1 + τ)2 Rk−1,n(τ, t)

≤ 4C4ω δk

(n∑

j=k


6

)1

(1 + τ)2,


where in the but-to-last step we used (En

Ω), (En

ρ ), Propositions 4.24 and 4.28, Con-dition (C5) and the same reasoning as in Step 5.

Summing up all contributions and inserting in (10.61) yields

(10.63)∥∥E(t, · )

∥∥Fλ∗

nt+µ∗n

≤ 4CF

[C4

ω

(C ′

0 +

n∑

k=1

δk

)(n∑

j=1

δj(2π(λj − λ∗n))

6

)+

n∑

k=1

δk

]∫ t

0

‖ρ[hn+1τ ]‖Fλ∗

nτ+µ∗n

dτ

(1 + τ)2.

b3. Main contribution: Now we consider σn,n+1, which we decompose as

σn,n+1t = σn,n+1

t,0 +n∑

k=1

σn,n+1t,k ,

where

σn,n+1t,0 (x) =

∫ t

0

∫F [hn+1]

(τ, x− v(t− τ), v

)· ∇vf

0(v) dv dτ,

σn,n+1t,k (x) =

∫ t

0

∫ (F [hn+1

τ ] · ∇v

(hk

τ Ωk−1t,τ

))(τ, x− v(t− τ), v

)dv dτ.

Note that their zero mode vanishes. For any k ≥ 1, we apply Theorem 6.5 (withM = 1) to get

∥∥σn,n+1t,k

∥∥Fλ∗

nt+µ∗n≤∫ t

0

Kn,hk

1 (t, τ)∥∥F [hn+1

τ ]∥∥Fν′n,γ dτ

+

∫ t

0

Kn,hk

0 (t, τ)∥∥F [hn+1

τ ]∥∥Fν′n,γ dτ,

where

• ν ′n = λ∗n(1 + b)

∣∣∣∣τ −bt

1 + b

∣∣∣∣+ µ′n

• Kn,hk

1 (t, τ) = sup0≤τ≤t

∥∥∥∇v

(hk

τ Ωk−1t,τ

)−⟨∇v

(hk

τ Ωk−1t,τ

)⟩∥∥∥Z

λk(1+b),µkτ−bt/(1+b)

1 + τ

Kn,k

1 ,

•Kn,k1 (t, τ) = (1+τ) d sup

ℓ 6=0, m6=0e−2π

“µk−µ∗

n2

”|m|

(e−2π(µ′

n−µ∗n)|ℓ−m|

1 + |ℓ−m|γ)e−2π

“λk−λ∗

n2

”|ℓ(t−τ)+mτ |

,

• Kn,hk

0 (t, τ) =

(sup

0≤τ≤t

∥∥∥∇v

⟨hk

τ Ωk−1t,τ

⟩∥∥∥Cλk(1+b);1

)Kn,k

0 ,


• Kn,k0 (t, τ) = d e

−2π“

λk−λ∗n

2

”(t−τ)

.

We assume

(10.64) µ′n = µ∗

n + η

(t− τ

1 + t

), η > 0 small,

and check that ν ′n ≤ λ∗nτ + µ∗n. Leaving apart the small-time case, we assume

τ ≥ bt/(1 + b), so that

ν ′n =(λ∗nτ + µ∗

n

)− B λ∗n (t− τ)

1 + t+ η

(t− τ

1 + t

),

which is indeed bounded above by λ∗nτ + µ∗n as soon as

(10.65) η ≤ B λ∗∞.

Then, with the notation (7.10),

(10.66) Kn,k1 (t, τ) ≤ K

(αn,k),γ1 (t, τ),

with

(10.67) αn,k = 2π min

µk − µ∗

n

2;λk − λ∗n

2; η

.

From the controls on hk (assumption (En

h)) we have

∥∥∥∇v

(hk

τ Ωk−1t,τ

)−⟨∇v

(hk

τ Ωk−1t,τ

)⟩∥∥∥Z

λk(1+b),µk;1

τ− bt1+b

≤∥∥∥∇v

(hk

τ Ωk−1t,τ

)∥∥∥Z

λk(1+b),µk;1

τ− bt1+b

≤ δk (1 + τ);

and∥∥∥⟨∇v

(hk

τ Ωk−1t,τ

)⟩∥∥∥Cλk(1+b);1

=∥∥∥⟨(∇v + τ∇x)

(hk

τ Ωk−1t,τ

)⟩∥∥∥Cλk(1+b);1

≤∥∥∥(∇v + τ∇x)

(hk

τ Ωk−1t,τ

)∥∥∥Z

λk(1+b);1

τ− bt1+b

≤ δk.


After controlling F [hn+1] by ρ[hn+1], we end up with

(10.68)∥∥σn,n+1

t,k

∥∥Fλ∗

nt+µ∗n≤ CF

∫ t

0

(n∑

k=1

δk K(αn,k),γ1 (t, τ)

)∥∥ρ[hn+1

τ ]∥∥Fλ∗

nτ+µ∗ndτ

+ CF

∫ t

0

(n∑

k=1

δk e−2π

“λk−λ∗

n2

”(t−τ)

)∥∥ρ[hn+1

τ ]∥∥Fλ∗

nτ+µ∗ndτ,

with αn,k defined by (10.67).

Substep c. Gathering all previous controls, we obtain the following integral inequalityfor ϕ = ρ[hn+1]:

(10.69)

∥∥∥∥ϕ(t, x) −∫ t

0

∫(∇W ∗ ϕ)(τ, x− v(t− τ)) · ∇vf

0(v) dv dτ

∥∥∥∥Fλ∗

nt+µ∗n

≤ An +

∫ t

0

[Kn

1 (t, τ) +Kn0 (t, τ) +

cn0(1 + τ)2

]‖ϕ(τ, · )‖Fλ∗

nτ+µ∗n ,

where, by (10.48), (10.60) and (10.63),

(10.70) An = supt≥0

∥∥σn,n(t, · )∥∥Fλ∗

nt+µ∗n≤ 2CF δ

2n

(π(λn − λ∗n))2,

Kn1 (t, τ) =

(CF

n∑

k=1

δk

)K

(αn),γ1 , αn = αn,n = 2π min

µn − µ∗

n

2;λn − λ∗n

2; η

,

Kn0 (t, τ) = CF

n∑

k=1

δk e−2π

“λk−λ∗

n2

”(t−τ)

,

cn0 = 3CF C4ω

(C ′

0 +

n∑

k=1

δk

) (n∑

k=1


6

)+

n∑

k=1

δk.

(We are cheating a bit when writing (10.69), because in fact one should take intoaccount small times separately; but this does not cause any difficulty.)

We easily estimate Kn0 :∫ t

0

Kn0 (t, τ) dτ ≤ CF

n∑

k=1

δkπ(λk − λ∗n)

,


∫ ∞

τ

Kn0 (t, τ) dt ≤ CF

n∑

k=1

δkπ(λk − λ∗n)

,

(∫ t

0

Kn0 (t, τ)2 dτ

)1/2

≤ CF

n∑

k=1

δk√2π(λk − λ∗n)

.

Let us assume that αn is smaller than α(γ) appearing in Theorem 7.7, and that

(10.71) (C7) 3CF C4ω

(C ′

0 +

n∑

k=1

δk + 1

) (n∑

k=1


6

)≤ 1

4,

(10.72) (C8) CF

n∑

k=1

δk√2π(λk − λ∗k)

≤ 1

2,

(10.73) (C9) CF

n∑

k=1

δkπ(λk − λ∗k)

≤ max

1

4; χ

,

(note that in these conditions we have strenghtened the inequalities by replacingλk − λ∗n by λk − λ∗k where χ > 0 is also defined by Theorem 7.7). Applying thattheorem with λ0 = λ, λ∗ = λ1, we deduce that for any ε ∈ (0, αn) and t ≥ 0,

(10.74) ‖ρn+1t ‖Fλ∗

nt+µ∗n ≤ C An

(1 + cn0 )2

√ε

eC cn0

(1 +

cnαn ε

)eC Tε,n eC cn (1+T 2

ε,n) eε t,

where

cn = 2CF

(n∑

k=1

δk

)

and

Tε,n = Cγ max

(c2n

α5n ε

2+γ

) 1γ−1

;

(cn

α2n ε

γ+ 12

) 1γ−1

;(cn0 )2/3

ε1/3

.

Pick up λ†n < λ∗n such that 2π(λ∗n − λ†n) ≤ αn, and choose ε = 2π(λ∗n − λ†n); recallingthat ρn+1(t, 0) = 0, and that our conditions imply an upper bound on cn and cn0 , wededuce the uniform control

‖ρn+1t ‖

Fλ†nt+µ∗

n≤ e−2π(λ∗

n−λ†n)t ‖ρn+1

t ‖Fλ∗nt+µ∗

n(10.75)

≤ C An

(1 +

1

αn (λ∗n − λ†n)3/2

)eC T 2

n ,


where

(10.76) Tn = C

(1

α5n (λ∗n − λ†n)2+γ

) 1γ−1

.

10.4.3. Step 7: estimate on F [hn+1]. As an immediate consequence of (10.2) and(10.75), we have

(10.77) supt≥0

∥∥F [ρn+1t ]

∥∥Fλ

†nt+µ∗

n,γ≤ CAn

(1 +

1

αn (λ∗n − λ†n)3/2

)eC T 2

n .

10.4.4. Step 8: estimate of hn+1 Ωn. In this step we shall use again the Vlasovequation. We rewrite (10.42) as

hn+1(τ,Xn

0,τ (x, v), Vn0,τ(x, v)

)=

∫ τ

0

Σn+1(s,Xn

0,s(x, v), Vn0,s(x, v)

)ds;

but now we compose with (Xnt,0, V

nt,0), where t ≥ τ is arbitrary. This gives

hn+1(τ,Xn

t,τ (x, v), Vnt,τ(x, v)

)=

∫ τ

0

Σn+1(s,Xn

t,s(x, v), Vnt,s(x, v)

)ds.

Then for any p ∈ [1, p] and λn < λ†n, using Propositions 4.19 and 4.24, and the

notation (10.44), we get

∥∥hn+1τ Ωn

t,τ

∥∥Z

(1+b)λn,µ∗

n;p

τ− bt1+b

=∥∥∥hn+1

τ (Xn

t,τ , Vnt,τ

)∥∥∥Z

(1+b)λn,µ∗

n;p

t− bt1+b

≤∫ τ

0

∥∥∥Σn+1(s,Xn

t,s, Vnt,s

)∥∥∥Z

(1+b)λn,µ∗

n;p

t− bt1+b

ds =

∫ τ

0

∥∥Σn+1(s,Ωnt,s)∥∥Z

(1+b)λn,µ∗

n;p

s− bt1+b

ds

≤∫ τ

0

∥∥Rn+1s,t

∥∥Z

(1+b)λn,µ∗

n

s− bt1+b

‖Gns,t‖Z(1+b)λ

n,µ∗n;p

s− bt1+b

ds

+

∫ τ

0

‖Rns,t‖Z(1+b)λ

n,µ∗n

s− bt1+b

‖Hns,t‖Z(1+b)λ

n,µ∗n;p

s− bt1+b

ds.

Then (proceeding as in Step 6 to check that the exponents lie in the appropriaterange)

‖Rn+1s,t ‖

Z(1+b)λ

n,µ∗n

s− bt1+b

≤ CF e−2π(λ†

n−λn)s ‖ρn+1

s ‖F νn(s)


and

‖Rns,t‖Z(1+b)λ,µ∗

n

s− bt1+b

≤ CF e−2π(λ†

n−λn)s ‖ρn

s‖F νn(s) ≤ CF e−2π(λ†

n−λn)s δn

withνn(s, t) := µ♯ when s ≤ bt/(1 + b)

νn(s, t) := λ†n s+ µ∗n when s ≥ bt/(1 + b).

On the other hand, from the induction assumption (En

h)-(En

h) (and again control

of composition via Proposition 4.25. . . ),

‖Hns,t‖Z(1+b)λ

n,µ∗n;p

s− bt1+b

≤ 2 (1 + s) δn

and

‖Gns,t‖Z(1+b)λ

n,µ∗n;p

s− bt1+b

≤ 2 (1 + s)

(n∑

k=1

δk

).

We deduce that

y(t, τ) :=∥∥hn+1

τ Ωnt,τ

∥∥Z

(1+b)λn,µ∗

n;p

τ− bt1+b

satisfies

y(t, τ) ≤ 2CF

(n∑

k=1

δk

) ∫ τ

0

e−2π(λ†n−λ

n)s ‖ρn+1s ‖F νn(s) (1 + s) ds

+ 2CF δ2n

∫ τ

0

e−2π(λ†n−λ

n)s (1 + s) ds;

so

(10.78) ∀ t ≥ τ ≥ 0,

∥∥hn+1τ Ωn

t,τ

∥∥Z

(1+b)λn,µ∗

n;p

τ− bt1+b

≤ 4CF max (∑n

k=1 δk) ; 1(2π(λ†n − λ

n))2

(δ2n + sup

s≥0‖ρn+1

s ‖F νn(s)

).


10.4.5. Step 9: Crude estimates on the derivatives of hn+1. Again we choose p ∈[1, p]. From the previous step and Proposition 4.27 we deduce, for any λ‡n such thatλ‡n < λ

n < λ‡n, and any µ‡n < µ∗

n,

(10.79)∥∥∥∇x

(hn+1

τ Ωnt,τ

)∥∥∥Z

λ‡n(1+b),µ

‡n;p

τ− bt1+b

+∥∥∥(∇v + τ∇x)

(hn+1

τ Ωnt,τ

)∥∥∥Z

λ‡n(1+b),µ

‡n;p

τ− bt1+b

≤ C(d)

min λn − λ‡n ; µ∗

n − µ‡n∥∥hn+1

τ Ωnt,τ

∥∥Z

λn(1+b),µ∗

n;p

τ− bt1+b

;

and(10.80)∥∥∥∇

(hn+1

τ Ωnt,τ

)∥∥∥Z

λ‡n(1+b),µ

‡n;p

τ− bt1+b

≤ C(d) (1 + τ)


n − µ‡n∥∥hn+1

τ Ωnt,τ

∥∥Z

λn(1+b),µ∗

n;p

τ− bt1+b

.

Similarly,(10.81)∥∥∥∇∇

(hn+1

τ Ωnt,τ

)∥∥∥Z

λ‡n(1+b),µ

‡n;p

τ− bt1+b

≤ C(d) (1 + τ)2


n − µ‡n2

∥∥hn+1τ Ωn

t,τ

∥∥Z

λn(1+b),µ∗

n;p

τ− bt1+b

.

10.4.6. Step 10: Chain-rule and refined estimates on derivatives of hn+1. FromStep 3 we have

(10.82)∥∥∇Ωn

t,τ

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

+∥∥(∇Ωn

t,τ )−1∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

≤ C(d)

and (via Proposition 4.27)

∥∥∥∇∇Ωnt,τ

∥∥∥Z

λ‡n(1+b),µ

‡n

τ− bt1+b

≤ C(d) (1 + τ)

min λ∗n − λ‡n ; µ∗n − µ‡

n∥∥∇Ωn

t,τ

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

(10.83)

≤ C(d) (1 + τ)

min λ∗n − λ‡n ; µ∗n − µ‡

n.

Combining these bounds with Step 9, Proposition 4.24 and the identities

(10.84)

(∇h) Ω = (∇Ω)−1 ∇(h Ω)

(∇∇h) Ω = (∇Ω)−2 ∇∇(h Ω) − (∇Ω)−1 ∇2Ω (∇Ω)−1(∇h Ω),


we get

∥∥∥(∇hn+1τ ) Ωn

t,τ

∥∥∥Z

λ‡n(1+b),µ

‡n;1

τ− bt1+b

≤ C(d)∥∥∥∇(hn+1

τ Ωnt,τ

)∥∥∥Z

λ‡n(1+b),µ

‡n;1

τ− bt1+b

(10.85)

≤ C(d) (1 + τ)

minλn − λ‡n ; µ∗

n − µ‡n

∥∥∥hn+1τ Ωn

t,τ

∥∥∥Z

λn(1+b),µ∗

n;1

τ− bt1+b

and

∥∥∥(∇2hn+1τ ) Ωn

t,τ

∥∥∥Z

λ‡n(1+b),µ

‡n;1

τ− bt1+b

(10.86)

≤ C(d)

∥∥∥∇2

(hn+1

τ Ωnt,τ

)∥∥∥Z

λ‡n(1+b),µ

‡n;1

τ− bt1+b

+∥∥∇2Ωn

t,τ

∥∥Z

λ‡n(1+b),µ

‡n;1

τ− bt1+b

∥∥(∇hn+1τ ) Ωn

t,τ

∥∥Z

λ‡n(1+b),µ

‡n;1

τ− bt1+b

≤ C(d) (1 + τ)2


n − µ‡n2

∥∥hn+1τ Ωn

t,τ

∥∥Z

λn(1+b),µ∗

n;p

τ− bt1+b

.

This gives us the bounds ‖(∇hn+1)Ωn‖ = O(1+τ), ‖(∇2hn+1)Ωn‖ = O((1+τ)2),which are optimal if one does not distinguish between the x and v variables. Weshall now refine these estimates. First we write

∇(hn+1τ Ωn

t,τ ) − (∇hn+1τ ) Ωn

t,τ = ∇(Ωnt,τ − Id ) ·

[(∇hn+1

τ ) Ωnt,τ

],

and we deduce (via Propositions 4.24 and 4.27)

∥∥∥∇(hn+1

τ Ωnt,τ

)− (∇hn+1

τ ) Ωnt,τ

∥∥∥Z

λ‡n(1+b),µ

‡n;p

τ− bt1+b

(10.87)

≤∥∥∇(Ωn

t,τ − Id )∥∥Z

λ‡n(1+b),µ

‡n

τ− bt1+b

∥∥(∇hn+1τ ) Ωn

t,τ

∥∥Z

λ‡n(1+b),µ

‡n;p

τ− bt1+b

≤ C(d)

(1 + τ


n − µ‡n

)2 ∥∥Ωnt,τ − Id

∥∥Z

λn(1+b),µ∗

n

τ− bt1+b

∥∥hn+1τ Ωn

t,τ

∥∥Z

λn(1+b),µ∗

n;p

τ− bt1+b

≤ C(d)C4ω


n − µ‡n2

(n∑

k=1


6

)(1 + τ)−2

∥∥hn+1τ Ωn

t,τ

∥∥Z

λn(1+b),µ∗

n;p

τ− bt1+b

.


(Note: Ωn − Id brings the time-decay, while hn+1 brings the smallness.)

This shows that (∇hn+1) Ωn ≃ ∇(hn+1 Ωn) as τ → ∞. In view of Step 9, thisalso implies the refined gradient estimates

(10.88)∥∥(∇xh

n+1τ ) Ωn

t,τ

∥∥Z

λ‡n,µ

‡n;p

τ− bt1+b

+∥∥∥((∇v + τ∇x)h

n+1τ

) Ωn

t,τ

∥∥∥Z

λ‡n,µ

‡n;p

τ− bt1+b

≤ C∥∥hn+1

τ Ωnt,τ

∥∥Z

λn(1+b),µ∗

n;p

τ− bt1+b

,

with

C = C(d)

[C4

ω


n − µ‡n2

(n∑

k=1


6

)+

1


n − µ‡n

].

10.4.7. Conclusion. Given λn+1 < λ∗n, µn+1 < µ∗n, we define

λn+1 = λ‡n, µn+1 = µ‡n,

and we impose

λ∗n − λ†n = λ†n − λn = λ

n − λ‡n =λ∗n − λn+1

3,

µ∗n − µ‡

n = µ∗n − µn+1.

Then from (10.75), (10.77), (10.78), (10.79), (10.86), (10.87) and (10.88) we see that

(En+1

ρ ), (En+1

ρ ), (En+1

h), (En+1

h) have all been established in the present subsection,

with(10.89)

δn+1 =C(d)CF (1 + CF ) (1 + C4

ω) eC T 2n

min λ∗n − λn+1 ; µ∗n − µn+19

max

(n∑

k=1

δk

); 1

(1 +

n∑

k=1


6

)δ2n.

10.5. Convergence of the scheme. For any n ≥ 1, we set

(10.90) λn − λ∗n = λ∗n − λn+1 = µn − µ∗n = µ∗

n − µn+1 =Λ

n2,

for some Λ > 0. By choosing Λ small enough, we can make sure that the conditions2π(λk−λ∗k) < 1, 2π(µk−µ∗

k) < 1 are satisfied for all k, as well as the other smallnessassumptions made throughout this section. Moreover, we have λk − λ∗k ≥ Λ/k2, soconditions (C1) to (C9) will be satisfied if

n∑

k=1

k12 δk ≤ Λ6 ω,n∑

j=k+1

j6 δj ≤ Λ3 ω

(1

k2− 1

n2

),


for some small explicit constant ω > 0, depending on the other constants appearingin the problem. Both conditions are satisfied if

(10.91)∞∑

k=1

k12 δk ≤ Λ6 ω.

Then from (10.76) we have Tn ≤ Cγ (n2/Λ)7+γγ−1 , so the induction relation on δn

allows

(10.92) δ1 ≤ C δ, δn+1 = C

(n2

Λ

)9

eC (n2/Λ)14+2γγ−1

δ2n.

To establish this relation we also assumed that δn is bounded below by CF ζn, theerror coming from the short-time iteration; but this follows easily by construction,since the constraints imposed on δn are much worse than those on ζn.

Having fixed Λ, we will check that for δ small enough, (10.92) implies both thefast convergence of (δk)k∈N, and the condition (10.91), which will justify a posteriorithe derivation of (10.92). (An easy induction is enough to turn this into a rigorousreasoning.)

For this we fix a ∈ (1, aI), 0 < z < zI < 1, and we check by induction

(10.93) ∀n ≥ 1, δn ≤ ∆ zan

.

If ∆ is given, (10.93) holds for n = 1 as soon as δ ≤ (∆/C) za. Then, to go fromstage n to stage n + 1, we should check that

C n18

Λ9eCn

28+4γγ−1 /Λ

14+2γγ−1

∆2 z2 an ≤ ∆ zan+1

;

this is true if1

∆≥ C

Λ9supn∈N

(n19 eC n

28+4γγ−1 /Λ

14+2γγ−1

z(2−a) an

).

Since a < 2, the supremum on the right-hand side is finite, and we just have to choose∆ small enough. Then, reducing ∆ further if necessary, we can ensure (10.91). Thisconcludes the proof.

Remark 10.1. This argument almost fully exploits the bi-exponential convergenceof the Netwon scheme: a convergence like, say, O(e−n1000

), would not be enough totreat values of γ which are close to 1. In Subsection 11.2 we shall present a morecumbersome approach which is less greedy in the convergence rate, but still needsconvergence like O(e−nα

) for α large enough.


11. Coulomb/Newton interaction

In this section we modify the scheme of Section 10 to treat the case γ = 1. Weprovide two different strategies. The first one is quite simple and will only comeclose to treat this case, since it will hold on (nearly) exponentially large times in theinverse of the perturbation size. The second one, somewhat more involved, will holdup to infinite times.

11.1. Estimates on exponentially large times. In this subsection we adapt theestimates of Section 10 to the case γ = 1, under the additional restriction that0 ≤ t ≤ A1/(δ(log δ)2) for some constant A > 1.

In the iterative scheme, the only place where we used γ > 1 (and not just γ ≥ 1)is in Step 6, when it comes to the echo response via Theorem 7.7. Now, in the caseγ = 1, the formula for Kn

1 should be

Kn1 (t, τ) =

n∑

k=1

δk K(αn,k),11 (t, τ),

with αn,k = 2π min (µk − µ∗n)/2 ; (λk − λ∗n)/2 ; η. By Theorem 7.7 (ii) this in-

duces, in addition to other well-behaved factors, an uncontrolled exponential growthO(eǫnt), with

ǫn = Γ

n∑

k=1

δkα3

n,k

;

in particular ǫn will remain bounded and O(δ) throughout the scheme.Let us replace (10.90) by

λn − λ∗n = λ∗n − λn+1 = µn − µ∗n = µ∗

n − µn+1 =Λ

n (log(e+ n))2,

where Λ > 0 is very small. (This is allowed since the series∑

1/(n(log(e + n))2)converges — the power 2 could of course be replaced by any r > 1.) Then during thefirst stages of the iteration we can absorb the O(eǫnt) factor by the loss of regularityif, say,

ǫn ≤ Λ

2n (log(e+ n))2.

Recalling that ǫn = O(δ), this is satisfied as soon as

(11.1) n ≤ N :=K

δ (log(1/δ))2,


whereK > 0 is a positive constant depending on the other parameters of the problembut of course not on δ. So during these first stages we get the same long-timeestimates as in Section 10.

For n > N we cannot rely on the loss of regularity any longer; at this stage theerror is about

δN ≤ C δaN

,

where 1 < a < 2. To get the bounds for larger values of n, we use impose a restrictionon the time-interval, say 0 ≤ t ≤ Tmax. Allowing a degradation of the rate δan

intoδan

with a < a, we see that the new factor eǫnTmax can be eaten up by the scheme if

eǫnTmax δ(a−a) an ≤ 1, ∀n ≥ N.

This is satisfied if

Tmax = O

(aN log 1

δ

δ

).

Recalling (11.1), we see that the latter condition holds true if

Tmax = O

(A

1δ(log δ)2

log 1δ

δ

)

for some well-chosen constant A > 1. Then we can complete the iteration, and endup with a bound like

‖ft − fi‖Zλ′,µ′

t≤ C δ ∀ t ∈ [0, Tmax],

where C is another constant independent of δ. The conclusion follows easily.

11.2. Mode-by-mode estimates. Now we shall change the estimates of Section10 a bit more in depth to treat arbitrarily large times for γ = 1. The main idea is towork mode by mode in the estimate of the spatial density, instead of looking directlyfor norm estimates.

Steps 1 to 5 remain the same, and the changes mainly occur in Step 6.Substep 6(a) is unchanged, but we only retain from that substep

(11.2) ∀ ℓ ∈ Zd, e2π(λ∗nt+µ∗

n)|ℓ|∣∣(σn,n

t )b(ℓ)∣∣ ≤ 2CF δ

2n

(π(λn − λ∗n))2.

Substep b is more deeply changed. Let µn < µ∗n.


• First, for each ℓ ∈ Zd, we have, by Propositions 4.34, 4.35 and the last part ofProposition 6.2,

e2π(λ∗nt+bµn)|ℓ|

∣∣E(t, ℓ)∣∣

≤∫ t

0

∑

m∈Zd

∥∥∥Pm

(F [hn+1


τ ])∥∥∥

Zλ∗

n(1+b),bµn

τ− bt1+b∥∥Pℓ−mG

nτ,t

∥∥Z

λ∗n(1+b),bµn;1

τ− bt1+b

dτ

≤∫ t

0

∑

m∈Zd

∑

m′∈Zd

∥∥∥∥Pm−m′

∫ 1

0

∇F [hn+1τ ]

(Id + θ(Ωn

t,τ − Id ))dθ

∥∥∥∥Z

λ∗n(1+b),bµn

τ− bt1+b∥∥Pm′

(Ωn

t,τ − Id)∥∥

Zλ∗

n(1+b),bµn

τ− bt1+b

∥∥Pℓ−mGnτ,t

∥∥Z

λ∗n(1+b),bµn;1

τ− bt1+b

dτ

≤∫ 1

0

∫ t

0

‖Gnτ,t‖Zλ∗

n(1+b),µ∗n;1

τ− bt1+b


∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

∑

m,m′∈Zd

e−2π(µ∗n−bµn)|ℓ−m|

e−2π(µ∗n−bµn)|m′|

∥∥∥Pm−m′

(∇F [hn+1

τ ] (Id + θ(Ωn

t,τ − Id )))∥∥∥

Zλ∗

n(1+b),bµn

τ− bt1+b

dτ dθ

≤∫ t

0

‖Gn‖Z

λ∗n(1+b),µ∗

n;1

τ− bt1+b


∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

∑

m, m′, q ∈Zd

e−2π(µ∗n−bµn)|ℓ−m| e−2π(µ∗

n−bµn)|m′|

e−2π(µ∗n−bµn)|m−m′−q|

∥∥∥Pq

(∇F [hn+1

τ ])∥∥∥

Fbνndτ,

where

νn = µn + λ∗n(1 + b)

∣∣∣∣τ −bt

1 + b

∣∣∣∣+∥∥ΩnXt,τ − x

∥∥Z

λ∗n(1+b),µ∗

n

τ− bt1+b

.

For α ≤ 1 we have

∑

m,m′∈Zd

e−2πα|ℓ−m| e−2πα|m′| e−2πα|m−m′−q| ≤ C(d)

αde−2π α

2|ℓ−q|,


we can argue as in Substep 6(b) of Section 10 to get

(11.3) e2π(λ∗nt+µ∗

n)|ℓ|∣∣E(t, ℓ)

∣∣ ≤ C

(µ∗n − µn)d

(C ′

0 +

n∑

k=1

δk

) (n∑

k=1


5

)

∑

q∈Zd

e−π(µ∗n−bµn)|ℓ−q|

∫ t

0

e2π(λ∗nτ+bµn)|q|

∣∣ρ[hn+1τ ]b(q)

∣∣ dτ

(1 + τ)2.

• Next, we use again Proposition 4.34 and simple estimates to bound E :


∣∣Eb(t, ℓ)

∣∣ ≤∫ t

0

‖Gn −Gn‖

Zλ∗

n(1+b),µ∗n;1

τ− bt1+b∑

m∈Zd

e−2π(µ∗n−bµn)|m|

∥∥Pℓ−m

(F [hn+1

τ ])∥∥

F bβndτ,

where βn = λ∗n(1 + b)|τ − bt/(1 + b)| + µn. Reasoning as in Substep 6(b) of Section10, we arrive at

(11.4) e2π(λ∗nt+bµn)|ℓ|

∣∣Eb(t, ℓ)

∣∣ ≤ C

(C ′

0 +

n∑

k=1

δk

)(n∑

j=1

δj(2π(λj − λ∗n))

6+

n∑

k=1

δk

)

∑

m∈Zd

e−2π(µ∗n−bµn)|m|

∫ t

0

e2π(λ∗nτ+bµn)|ℓ−m|

∣∣ρ[hn+1τ ]b(ℓ−m)

∣∣ dτ

(1 + τ)2.

• Then we consider the “main contribution” σn,n+1, which we decompose as inSection 10:

σn,n+1t = σn,n+1

t,0 +

n∑

k=1

σn,n+1t,k ,

and we write for k ≥ 1:

e2π(λ∗nt+bµn)

∣∣(σn,n+1t,k

)b(ℓ)∣∣ ≤

∑

m∈Zd

∫ t

0

Kn,hk

ℓ,m (t, τ)∥∥Pℓ−m

(F [hn+1

τ ])∥∥

Fν′n,γ dτ

+

∫ t

0

Kn,hk

0 (t, τ)∥∥Pℓ

(F [hn+1

τ ])∥∥

Fν′n,γ dτ,


where

ν ′n = λ∗n(1 + b)

∣∣∣∣τ −bt

1 + b

∣∣∣∣+ µ′n,

Kn,hk

ℓ,m (t, τ) = sup0≤τ≤t

sup0≤τ≤t

∥∥∥∇v

(hk

τ Ωk−1t,τ

)−⟨∇v

(hk

τ Ωk−1t,τ

)⟩∥∥∥Z

λk(1+b),µkτ−bt/(1+b)

1 + τ

Kn,k

ℓ,m,

Kn,kℓ,m(t, τ) = (1 + τ) e

−2π“

µk−bµn2

”|m|

(e−2π(µ′

n−bµn)|ℓ−m|

1 + |ℓ−m|γ)e−2π

“λk−λ∗

n2

”|ℓ(t−τ)+mτ |

,

and the formula for Kn,hk

0 is unchanged with respect to Section 10.Assuming µ′

n = µn+η (t−τ)/(1+t) and reasoning as in Substep 6(b) of Section 10,we end up with the following estimate on the “main term”:

(11.5) e2π(λ∗nt+bµn)|ℓ|

∣∣(σn,n+1t,k

)b(ℓ)∣∣

≤ C∑

m∈Zd

∫ t

0

(n∑

k=1

δk K(αn,k),γ

ℓ,m (t, τ)

)e2π(λ∗

nτ+bµn)|ℓ−m|∣∣ρ[hn+1

τ ]b(ℓ−m)∣∣ dτ

+ C∑

m∈Zd

∫ t

0

(n∑

k=1

δke−2π

“λk−λ∗

n2

”(t−τ)

)e2π(λ∗

nτ+bµn)|ℓ|∣∣ρ[hn+1

τ ]b(ℓ−m)∣∣ dτ.

Then Substep 6(c) becomes, with Φ(ℓ, t) = ρ[hn+1τ ]b(ℓ),


∣∣∣∣Φ(ℓ, t) −∫ t

0

K0(ℓ, t− τ) Φ(ℓ, τ) dτ

∣∣∣∣ ≤C δ2

n

(λn − λ∗n)2

+∑

m∈Zd

∫ t

0

[Kn

ℓ,m(t, τ) +cnm

(1 + τ)2

]e2π(λ∗

nτ+bµn)|ℓ−m| |Φ(ℓ−m, τ)| dτ

+

∫ t

0

Kn0 (t, τ) e2π(λ∗

nτ+bµn)|ℓ| |Φ(ℓ, τ)| dτ,

with

Knℓ,m = C

n∑

k=1

δk K(bαn),γℓ,m , αn = 2π min

µn − µn

2;λn − λ∗n

2; η

,

Kn0 (t, τ) = C

n∑

k=1

δk e−2π

“λk−λ∗

n2

”(t−τ)

,


cnm =C

(µ∗n − µn)d

(C ′

0 +n∑

k=1

δk

) (1 +

n∑

k=1


6

)e−π(µ∗

n−bµn)|m|.

Note that

∑

m∈Zd

cnm+

(∑

m∈Zd

(cnm)2

)1/2

≤ C

(µ∗n − µn)2d

(C ′

0 +n∑

k=1

δk

)(1 +

n∑

k=1


6

).

Then we can apply Theorem 7.12 and deduce (taking already into account, forthe sake of lisibility of the formula, that

∑δk and

∑δk/(λk − λ∗k) are uniformly

bounded)


∣∣(ρn+1t )b(ℓ)

∣∣ ≤ Cδ2n

(λn − λ∗n)2 αn ε3/2 (µ∗n − µn)2d

exp

(C (1 + T 2

ε,n)

(µ∗n − µn)2d

)eεt,

where

Tε,n = C max

(1

α3+2dn εγ+2

) 1γ

;

(1

αdn ε

γ+ 12

) 1

γ− 12

;

((∑

m cnm)2

ε

) 13

.

If λ†n < λ∗n and µ†n < µn are chosen as before and ε = 2π(λ∗n − λ†), this implies a

uniform bound on

‖ρn+1t ‖

Fλ†nt+µ

†n≤ C

(µn − µ†n)d

supℓ∈Zd


∣∣(ρn+1t )b(ℓ)

∣∣

obtained with the formula above with

Tε,n = Tn = C max

1

λ∗n − λ†n,

1

λn − λ∗n,

1

µn − µ∗n

,1

µ∗n − µn

max 5+γ+2dγ

;d+γ+1/2

γ−1/2; 4d+1

3 .

Then Steps 7 to 10 of the iteration can be repeated with the only modificationthat µ∗

n is replaced by µ†n.

The convergence (Subsection 10.5) works just the same, except that now we needmore intermediate regularity indices µn:

µn+1 = µ‡n < µ†

n < µn < µ∗n;

the obvious choice being to let µ†n − µ‡

n = µn − µ†n = µ∗

n − µn.


Choosing λn − λn+1 and µn − µn+1 of the order of Λ/n2, we arrive in the end atthe induction

δn+1 ≤ C

(n2

Λ

)9+6d

eC

“n2

Λ

”ξ(d,γ)

δ2n

with

(11.6) ξ(d, γ) := 2d+ 2 max

5 + γ + 2d

γ;d+ γ + 1/2

γ − 1/2;

4d+ 1

3

.

Then the convergence of the scheme (and a posteriori justification of all the as-sumptions) is done exactly as in Section 10.

12. Convergence in large time

In this section we prove Theorem 2.6 as a simple consequence of the uniformbounds established in Sections 10 and 11.

So let f 0, L,W satisfy the assumptions of Theorem 2.6. To simplify notation weassume L = 1.

The second part of Assumption (2.14) precisely means that f 0 ∈ Cλ;1. We shallactually assume a slightly more precise condition, namely that for some p ∈ [1,∞],

(12.1)∑

n∈Nd0

λn

n!‖∇n

vf0‖Lp(Rd) ≤ C0 < +∞, ∀ p ∈ [1, p].

(It is sufficient to take p = 1 to get Theorem 2.6; but if this bound is available forsome p > 1 then it will be propagated by the iteration scheme, and will result inmore precise bounds.) Then we pick up λ ∈ (0, λ), µ ∈ (0, µ), β > 0, β ′ ∈ (0, β). Bysymmetry, we only consider nonnegative times.

If fi is an initial datum satisfying the smallness condition (2.15), then by Theorem4.20, we have a smallness estimate on ‖fi−f 0‖Zλ′,µ′;p for all p ∈ [1, p], λ′ < λ, µ′ < µ.Then, as in Subsection 4.12 we can estimate the solution h1 to the linearized equation

(12.2)

∂th

1 + v · ∇xh1 + F [h1] · ∇vf

0 = 0

h1(0, · ) = fi − f 0,

and we recover uniform bounds in Zbλ,bµ;p spaces, for any λ ∈ (λ, λ), µ ∈ (µ, µ). Moreprecisely,

(12.3) supt≥0

‖ρ[h1t ]‖Fbλt+bµ + sup

t≥0‖h1(t, · )‖

Zbλ,bµ;pt

≤ C δ,


with C = C(d, λ′, λ, µ′, µ,W, f 0) (this is of course assuming ε in Theorem 2.6 to besmall enough).

We now set λ1 = λ′, and we run the iterative scheme of Sections 9–10–11 for alln ≥ 2. If ε is small enough, up to slightly lowering λ1, we may choose all parametersin such a way that

λk, λ∗k −−−→

k→∞λ∞ > λ, µk, µ

∗k −−−→

k→∞µ∞ > µ;

then we pick up B > 0 such that

µ∞ − λ∞(1 +B)B ≥ µ′∞ > µ,

and we let b(t) = B/(1 + t).As a result of the scheme, we have, for all k ≥ 2,

(12.4) supt≥τ≥0

∥∥hkτ Ωk−1

t,τ

∥∥Z

λ∞(1+b),µ∞;1

τ− bt1+b

≤ δk,

where∑∞

k=2 δk ≤ C δ and Ωk is the deflection associated to the force field generatedby h1 + . . .+ hk. Choosing t = τ in (12.4) yields

supt≥0

‖hkt ‖Zλ∞(1+B),µ∞;1

t− Bt1+B+t

≤ δk.

By Proposition 4.17, this implies

supt≥0

‖hkt ‖Zλ∞(1+B),µ∞−λ∞(1+B)B;1

t≤ δk.

In particular, we have a uniform estimate on hkt in Zλ∞,µ′

∞;1t . Summing up over k

yields for f = f 0 +∑

k≥1 hk the estimate

(12.5) supt≥0

∥∥f(t, · ) − f 0∥∥Z

λ∞,µ′∞;1

t

≤ C δ.

Passing to the limit in the Newton scheme, one shows that f solves the nonlinearVlasov equation with initial datum fi. (Once again we do not check details; tobe rigorous one would need to establish moment estimates, locally in time, beforepassing to the limit.) This implies in particular that f stays nonnegative at all times.

Applying Theorem 4.20 again, we deduce from (12.5)

supt≥0

∥∥f(t, · ) − f 0∥∥Y

λ,µ

t

≤ C δ;

or equivalently, with the notation used in Theorem 2.6,

(12.6) supt≥0

∥∥f(t, x− vt, v) − f 0(v)∥∥

λ,µ≤ C δ.


Moreover, ρ =∫f dv satisfies similarly

supt≥0

‖ρ(t, · )‖Fλ∞t+µ∞ ≤ C δ.

It follows that |ρ(t, k)| ≤ C δ e−2πλ∞|k|t e−2πµ∞|k|, for any k 6= 0. On the one hand,by Sobolev embedding, we deduce that for any r ∈ N,

‖ρ(t, · ) − 〈ρ〉‖Cr(Td) ≤ Cr δ e−2πλ′t;

on the other hand, multiplying ρ by the Fourier transform of ∇W , we see that theforce F = F [f ] satisfies

(12.7) ∀ t ≥ 0, ∀ k ∈ Zd, |F (t, k)| ≤ C δ e−2πλ′|k|t e−2πµ′|k|,

for some λ′ > λ, µ′ > µ.

Now, from (12.6) we have, for any (k, η) ∈ Zd × Rd, and any t ≥ 0,

(12.8)∣∣∣f(t, k, η + kt) − f 0(η)

∣∣∣ ≤ C δ e−2πµ′|k| e−2πλ′|η|;

so

(12.9) |f(t, k, η)| ≤∣∣f 0(η + kt)

∣∣ + C δ e−2πµ′|k| e−2πλ′|η+kt|.

In particular, for any k 6= 0, and any η ∈ Rd,

(12.10) f(t, k, η) = O(e−2πλ′t).

Thus f is asymptotically close (in the weak topology) to its spatial average g =〈f〉 =

∫f dx. Taking k = 0 in (12.8) shows that, for any η ∈ Rd,

(12.11) |g(t, η) − f 0(η)| ≤ C δ e−2πλ′|η|.

Also, from the nonlinear Vlasov equation, for any η ∈ Rd we have

g(t, η) = fi(0, η) −∫ t

0

∫

TdL

∫

Rd

F (τ, x) · ∇vf(τ, x, v) e−2iπη·v dv dx dτ

= fi(0, η) − 2iπ∑

ℓ∈Zd

∫ t

0

F (τ, ℓ) · η f(τ,−ℓ, η) dτ.

Using the bounds (12.7) and (12.10), it is easily shown that the above time-integralconverges exponentially fast as t → ∞, with rate O(e−λ′′t) for any λ′′ < λ′, to itslimit

(12.12) g∞(η) = fi(0, η) − 2iπ∑

ℓ∈Zd

∫ ∞

0

F (τ, ℓ) · η f(τ,−ℓ, η) dτ.


By passing to the limit in (12.11) we see that

|g∞(η) − f 0(η)| ≤ C δ e−2πλ′|η|,

and this concludes the proof of Theorem 2.6.

13. Non-analytic perturbations

Although the vast majority of studies of Landau damping assume that the per-turbation is analytic, it is natural to ask whether this condition can be relaxed. Aswe noticed in Remark 3.5, this is the case for the linear problem. As for nonlinearLandau damping, once the analogy with KAM theory has been identified, it is any-body’s guess that the answer might come from a Moser-type argument. However,this is not so simple, because the “loss of convergence” in our argument is muchmore severe than the “loss of regularity” which Moser’s scheme allows to overcome.

For instance, the second-order correction h2 satisfies

∂th2 + v · ∇xh

2 + F [f 1] · ∇vh2 + F [h2] · ∇vf

1 = −F [h1] · ∇vh1.

The action of F [f 1] is to curve trajectories, which does not help in our estimates.Discarding this effect and solving by Duhamel’s formula and Fourier transform, weobtain, with S = −F [h1] · ∇vh

1, ρ2 =∫h2 dv,

(13.1) ρ2(t, k) ≃∫ t

0

K0(t− τ, k) ρ2(τ, k) dτ

+2iπ

∫ t

0

∑

ℓ

(k−ℓ) W (k−ℓ) ρ2(τ, k−ℓ) ]∇vh1(τ, ℓ, k(t−τ)

)dτ+

∫ t

0

S(τ, k, k(t−τ)

)dτ.

(The term with K0 includes the contribution of ∇vf0.)

Our regularity/decay estimates on h1 will never be better than those on the so-lution of the free transport equation, i.e., hi(x − vt, v), where hi = fi − f 0. Let usforget about the effect of K0 in (13.1), replace the contribution of S by a decaying

term A(kt). Let us choose d = 1 and assume hi(ℓ, · ) = 0 if ℓ 6= ±1. For k > 0 letus use the long-time approximation

hi(−1, k(t− τ) − τ) 1[0,t](τ) dτ ≃ c

k + 1δ kt

k+1, c =

∫hi(−1, s) ds = hi(−1, 0);


note that c 6= 0 in general. Plugging all these simplifications in (13.1) and choosing

W (k) = 1/|k|1+γ suggests the a priori simpler equation

(13.2) ϕ(t, k) = A(kt) +ckt

(k + 1)γ+2ϕ

(kt

k + 1, k + 1

).

Replacing ϕ(t, k) by ϕ(t, k)/A(kt) reduces to A = 1, and then we can solve thisequation by power series as in Subsection 7.1.3, obtaining

(13.3) ϕ(t, k) ≃ A(kt) e(ckt)1

γ+2.

With a polynomial deterioration of the rate we could use a regularization argument,but the fractional exponential is much worse.

However, our bounds are still good enough to establish decay for Gevrey pertur-bations. Let us agree that a function f = f(x, v) lies in the Gevrey class Gν , ν ≥ 1,

if |f(k, η)| = O(exp(−c|(k, η)|1/ν)

)for some c > 0; in particular G1 means analytic.

(An alternative convention would be to require the nth derivative to be O(n!ν).) Aswe shall explain, we can still get nonlinear Landau damping if the initial datum fi

lies in Gν for ν close enough to 1. Although this is still quite demanding, it alreadyshows that nonlinear Landau damping is not tied to analyticity or quasi-analyticity,and holds for a large class of compactly supported perturbations.

Theorem 13.1. Let λ > 0. Let f 0 = f 0(v) ≥ 0 be an analytic homogeneous profilesuch that ∑

n∈Nd0

λn

n!‖∇n

vf0‖L1(Rd) < +∞,

and let W = W (x) satisfy |W (k)| = O(1/|k|2), such that Condition (L) from Sub-section 2.2 holds. Let ν ∈ (1, 1 + θ) with θ = 1/ξ(d, γ), where ξ was defined in(11.6). Let β > 0 and let α < 1/ν. Then there is ε > 0 such that if

δ := supk,η

(∣∣(fi − f 0)(k, η)∣∣ eλ|η|1/ν

eλ|k|1/ν)

+

∫∫ ∣∣(fi − f 0)(x, v)∣∣ eβ|v| dv dx ≤ ε,

then as t → +∞ the solution f = f(t, x, v) of the nonlinear Vlasov equation onTd × Rd with interaction potential W and initial datum fi satisfies, for all r ∈ N,

∀ (k, η),∣∣∣f(t, k, η) − f∞(η)

∣∣∣ = O(δ e−ctα

);

‖F (t, · )‖Cr(Td) = O(δ e−ctα

)

for some c > 0 and some homogeneous Gevrey profile f∞, where F stands for theself-consistent force.


Remark 13.2. In view of (13.3), one may hope that the result remains true forθ = 2. Proving this would require much more precise estimates, including amongother things a qualitative improvement of the constants in Theorem 4.20 (recallRemark 4.23).

Remark 13.3. One could also relax the analyticity of f 0, but there is little incentiveto do so.

Sketch of proof of Theorem 13.1. We first decompose hi = fi − f 0, using truncationby a smooth partition of unity in Fourier space:

hi =∑

n≥0

F−1(hi χn

)≡∑

n≥0

hni ,

where F is the Fourier transform. Each bump function χn should be localized aroundthe domain (in Fourier space)

Dn =nK ≤ |(k, η)| ≤ (n+ 1)K

,

for some exponent K > 1; but at the same time F−1(χn) should be exponentiallydecreasing in v. To achieve this, we let

χn = 1Dn ∗ γ, γ(η) = e−π|η|2 .

Then F−1(χn) = F−1(1Dn) γ has Gaussian decay, independently of n; so there is auniform bound on

∫∫|hn

i (x, v)| eβ|v| dv dx, for some β > 0.On the other hand, if (k, η) ∈ Dn and (k′, η′) /∈ (Dn−1∪Dn∪Dn+1), then |k−k′|+

|η − η′| ≥ c nK−1 for some c > 0; from this one obtains, after simple computations,∣∣χn(k, η)

∣∣ ≤ 1(n−1)K≤|(k,η)|≤(n+2)K + C e−c n2(K−1)

e−c (|k|2+|η|2).

So (with constants C and c changing from line to line)

∣∣hni (k, η)

∣∣ ≤ C e−λ|k|1ν e−λ|η|

1ν 1(n−1)K≤|(k,η)|≤(n+2)K + C e−c n2(K−1)

e−c(|k|+|η|)

≤ C maxe−

λ2

(n−1)Kν , e−c n2(K−1)

e−λn(|k|+|η|),

where

λn ∼ λ

2(n + 2)−

(1− 1

ν

)K .

If K ≥ 2 then 2(K − 1) > K/ν; so ‖hni ‖Yλn,λn ≤ C e−

λ2

nK/ν. Then we may apply

Theorem 4.20 to get a bound on hni in the space Zbλn,bλn;1 with λn = λn/2, at the


price of a constant exp(C (n+ 2)(1−1/ν)K). Assuming Kν > (1− 1/ν)K, i.e., ν < 2,we end up with

(13.4) ‖hni ‖Zbλn,bλn ;1

= O(e−cnK/ν)

, λn =λn

2.

Then we run the iteration scheme of Section 8 with the following modifications:(1) instead of hn(0, · ) = 0, choose hn(0, · ) = hn

i , and (2) choose regularity indices

λn ∼ λn which go to zero as n goes to infinity. This generates an additional error

term of size O(exp(−c nK

ν )), and imposes that λn −λn+1 be of order n−

[(1− 1

ν

)K+1].

When we apply the bilinear estimates from Section 6, we can take λ − λ to be of

the same order; so α = αn and ε = εn can be chosen proportional to n−[(

1− 1ν

)K+1].

Then the large constants coming from the time-response will be, as in Section 11,of order nq ecnr

, with q ∈ N and r = [(1 − 1/ν)K + 1]ξ, and the scheme will stillconverge like O(e−cns

) for any s < K/ν, provided that K/ν > r, i.e.,

(ν − 1) +ν

K<

1

ξ.

The rest of the argument is similar to what we did in Sections 10 to 12. In theend the decay rate of any nonzero mode of the spatial density ρ is controlled by

∑

n

e−cns

e−λnt ≤(∑

n

e−cns

)sup

n

[exp(−cns) exp

(−c n−

(1− 1

ν

)t)]

≤ C exp(−c ts/K),

and the result follows since s/K is arbitrarily close to 1/ν.

Remark 13.4. An alternative approach to Gevrey regularity consists in rewritingthe whole proof with the help of Gevrey norms such as

‖f‖Cλν

=∑

n∈N

λn ‖f (n)‖∞n!ν

, ‖f‖Fλν

=∑

k∈Z

e2πλ|k|1/ν |f(k)|,

which satisfy the algebra property for any ν ≥ 1. Then one can hybridize thesenorms, rewrite the time-response in this setting, estimate fractional exponentialmoments of the kernel, etc. The only part which does not seem to adapt to thisstrategy is Section 9 where the analyticity is crucially used for the local result.

Remark 13.5. In a more general Cr context, we do not know whether decay holdsfor the nonlinear Vlasov–Poisson equation. Speculations about this issue can be


found in [53] where it is shown that (unlike in the linearized case) one needs morethan one derivative on the perturbation. As a first step in this direction, we mentionthat our methods imply a bound like O(δ/(1+t)r−r) for times t = O(1/δ), where r isa constant and r > r, as soon as the initial perturbation has norm δ in a functionalspace Wr involving r derivatives in a certain sense. The reason why this is nontrivialis that the natural time scale for nonlinear effects in the Vlasov–Poisson equationis not O(1/δ), but O(1/

√δ), as predicted by O’Neil [73] and very well checked in

numerical simulations [59].17 Let us sketch the argument in a few lines. Assumethat (for some positive constants c, C)

(13.5) hi =∑

n

hni , ‖hn

i ‖Zλn,λn;1 ≤ Cn

2rn, λn =

c n

2n.

Then we may run the Newton scheme again choosing αn ∼ c n/2n, cn = O(δ 2−(r−r1)n)and εn = c′ δ. Over a time-interval of length O(1/δ), Theorem 7.7(ii) only yields amultiplicative constant O(ec δ t/α9

n) = O(210n). In the end, after Sobolev injectionagain, we recover a time-decay on the force F like

δ∑

n

2nr2 2−nr e−λnt ≤ C δ supn

(2−n(r−r3) e−λnt

)≤ C δ

(1 + t)r−r4,

as desired. Equation (13.5) means that hi is of size O(δ) in a functional space Wr

whose definition is close to the Littlewood–Paley characterization of a Sobolev spacewith r derivatives. In fact, if the conjecture formulated in Remark 4.23 holds true,then it can be shown that Wr contains all functions in the Sobolev space W r+r0,2

satisfying an adequate moment condition, for some constant r0.

14. Expansions and counterexamples

A most important consequence of the proof of Theorem 2.6 is that the asymptoticbehavior of the solution of the nonlinear Vlasov equation can in principle be deter-mined at arbitrary precision as the size of the perturbation goes to 0. Indeed, if wedefine gk

∞(v) as the large-time limit of hk (say in positive time), then ‖gk‖ = O(δk),so f 0 + g1

∞ + . . . + gn∞ converges very fast to f∞. In other words, to investigate

the properties of the time-asymptotics of the system, we may freely exchange the

17Passing from O(1/√

δ) to O(1/δ) is arguably an infinite-dimensional counterpart of Laplace’saveraging principle, which yields stability for certain Hamiltonian systems over time intervalsO(1/δ2) rather than O(1/δ).


limits t → ∞ and δ → 0, perform expansions, etc. This at once puts on rigor-ous grounds many asymptotic expansions used by various authors — who so farimplicitly postulated the possibility of this exchange.

With this in mind, let us estimate the first corrections to the linearized theory, inthe regime of a very small perturbation and small interaction strength (which canbe achieved by a proper scaling of physical quantities). We shall work in dimensiond = 1 and in a periodic box of length L = 1.

14.1. Simple excitation. For a start, let us consider the case where the perturba-tion affects only the first spatial frequency. We let

• f 0(v) = e−π v2: the homogeneous (Maxwellian) distribution;

• ε ρi(x) = ε cos(2πx): the initial space density perturbation;

• ε ρi(x) θ(v): the initial perturbation of the distribution function; we denote byϕ the Fourier transform of θ;

• αW : the interaction potential, with W (−x) = W (x). We do not specify itsform, but it should satisfy the assumptions in Theorem 2.6.

We work in the asymptotic regime ε → 0, α → 0. The parameter ε measuresthe size of the perturbation, while α measures the strength of the interaction; afterdimensional change, if W is an inverse power, α can be thought of as an inversepower of the ratio (Debye length)/(perturbation wavelength). We will not writenorms explicitly, but all our computations can be made in the norms introduced inSection 4, with small losses in the regularity indices — as we have done in all thispaper.

The first-order correction h1 = O(ε) to f 0 is provided by the solution of thelinearized equation (3.3), here taking the form

∂th1 + v · ∇xh

1 + F [h1] · ∇vf0 = 0,

with initial datum h1(0, · ) = hi := fi − f 0. As in Section 3 we get a closed equationfor the associated density ρ[h1]:

ρ[h1](t, k) = hi(k, kt) − 4π2 α W (k)

∫ t

0

ρ[h1](τ, k) e−π(k(t−τ))2 (t− τ) k2 dτ.


It follows that ρ[h1](t, k) = 0 for k 6= ±1, so the behavior of ρ[h1] is entirely deter-mined by u1(t) = ρ[h1](t, 1) and u−1(t) = ρ[h1](t,−1), which satisfy

u1(t) =ε

2ϕ(t) − 4π2α W (1)

∫ t

0

u1(τ) e−π(t−τ)2 (t− τ) dτ(14.1)

=ε

2

[ϕ(t) +O(α)

].

(This equation can be solved explicitly [11, eq. 6], but we only need the expansion.)Similarly,

u−1(t) =ε

2ϕ(−t) − 4π2α W (1)

∫ t

0

u−1(τ) e−π(t−τ)2 (t− τ) dτ(14.2)

=ε

2

[ϕ(−t) +O(α)

].

The corresponding force, in Fourier transform, is given by F 1(t, 1) = −2iπα W (1) u1(t)

and F 1(t,−1) = 2iπα W (1) u−1(t).From this we also deduce the Fourier transform of h1 itself:

(14.3)

h1(t, k, η) = hi(k, η+kt)−4π2α W (k)

∫ t

0

ρ[h1](τ, k) e−π(η+k(t−τ))2(η+k(t−τ)

)·k dτ ;

this is 0 if k 6= ±1, while

h1(t, 1, η) =ε

2ϕ(η + t) − 4π2α W (1)

∫ t

0

u1(τ) e−π(η+(t−τ))2

(η + (t− τ)

)dτ(14.4)

=ε

2

[ϕ(η + t) +O(α)

],

h1(t,−1, η) =ε

2ϕ(η − t) + 4π2α W (1)

∫ t

0

u−1(τ) e−π(η−(t−τ))2

(η − (t− τ)

)dτ

(14.5)

=ε

2

[ϕ(η − t) +O(α)

].

To get the next order correction, we solve, as in Section 10,

∂th2 + v · ∇xh

2 + F [h1] · ∇vh2 + F [h2] · (∇vf

0 + ∇vh1) = −F [h1] · ∇vh

1,

with zero initial datum. Since h2 = O(ε2), we may neglect the terms F [h1] · ∇vh2

and F [h2] · ∇vh1 which are both O(αε3). So it is sufficient to solve

(14.6) ∂th′2 + v · ∇xh

′2 + F [h′2] · ∇vf

0 = −F [h1] · ∇vh1


with vanishing initial datum. As t → ∞, we know that the solution h′2(t, x, v) isasymptotically close to its spatial average 〈h′2〉 =

∫h′2 dx. Taking the integral over

Td in (14.6) yields

∂t〈h′2〉 = −⟨F [h1] · ∇vh

1⟩.

Since h1 converges to 〈hi〉, the deviation of f to 〈fi〉 is given, at order ε2, by

g(v) = −∫ +∞

0

⟨F [h1] · ∇vh

1⟩(t, v) dt

= −∫ +∞

0

∑

k∈Z

F [h1](t,−k) · ∇vh1(t, k, v) dt.

Applying the Fourier transform and using (14.1)-(14.2)-(14.4)-(14.5), we deduce

g(η) = −∫ +∞

0

∑

k∈Z

F [h1](t,−k) · ∇vh1(t, k, η) dt

= −∫ +∞

0

F [h1](t,−1) (2iπη) h1(t, 1, η) dt

−∫ +∞

0

F [h1](t, 1) (2iπη) h1(t,−1, η) dt

= π2ε2α W (1) η

(∫ +∞

0

ϕ(−t)ϕ(η + t) dt−∫ +∞

0

ϕ(t)ϕ(η − t) dt+O(α)

)

= −π2ε2α W (1) η

(∫ +∞

−∞

ϕ(t)ϕ(η − t) sign (t) dt+O(α)

).

Summarizing:(14.7)

limt→∞

f(t, k, η) = 0 if k 6= 0

limt→∞

f(t, 0, η) = fi(t, 0, η) − ε2 α(π2 W (1)

)η

(∫ +∞

−∞

ϕ(t)ϕ(η − t) sign (t) dt+O(α)

).

Since ϕ is an arbitrary analytic profile, this simple calculation already shows thatthe asymptotic profile is not necessarily the spatial mean of the initial datum.


Assuming ε ≪ α, higher order expansions in α can be obtained by bootstrap onthe equations (14.1)-(14.2)-(14.4)-(14.5): for instance,

limt→∞

f(t, 0, η) = fi(0, η) − ε2 α(π2 W (1)

)η

∫ +∞

−∞

ϕ(t)ϕ(η − t) sign (t) dt

− ε2 α2(2π2W (1)

)2η

∫ ∞

0

∫ t

0

(ϕ(η + t)ϕ(−τ) − ϕ(η − t)ϕ(τ)

)e−π(t−τ)2 (t− τ)

+ ϕ(τ)ϕ(−t) e−π(η+(t−τ))2(η + (t− τ)

)

+ ϕ(−τ)ϕ(t) e−π(η−(t−τ))2(η − (t− τ)

)dτ +O(ε2α3).

What about the limit in negative time? Reversing time is equivalent to changingf(t, x, v) into f(t, x,−v) and letting time go forward. So we define S(v) := −v,T (ϕ)(η) := ε2 α π2 W (1) η

∫ +∞

−∞ϕ(t)ϕ(η − t) sign (t) dt; then T (ϕ S) = T (ϕ) S,

which means that the solutions constructed above are always homoclinic at orderO(ε2α). The same is true for the more precise expansions at order O(ε2α2), and infact it can be checked that the whole distribution f 2 is homoclinic; in other words,f is homoclinic up to possible corrections of order O(ε4). To exhibit heteroclinicdeviations, we shall consider more general perturbations.

14.2. General perturbation. Let us now consider a “general” initial datum fi(x, v)close to f 0(v), and expand the solution f . We write ε ϕk(η) = (fi − f 0) f (k, η) andρm = ρ[hm]. The interaction potential is assumed to be of the form αW with α≪ 1and W (x) = W (−x). The first equations of the Newton scheme are

(14.8) ρ1(t, k) = ε ϕk(kt) − 4π2α W (k)

∫ t

0

ρ1(τ, k) f 0(k(t− τ)

)|k|2 (t− τ) dτ,

(14.9)

h1(t, k, η) = ε ϕk(η+kt)−4π2α W (k)

∫ t

0

ρ1(τ, k) f 0(η+k(t−τ)

)k ·(η+k(t−τ)

)dτ,


h2(t, k, η) = −4π2α W (k)

∫ t

0

ρ2(τ, k) f 0(η + k(t− τ)

)k ·(η + k(t− τ)

)dτ

(14.10)

− 4π2α

∫ t

0

∑

ℓ

W (ℓ) ρ1(τ, ℓ) h1(τ, k − ℓ, η + k(t− τ)

)ℓ ·(η + k(t− τ)

)dτ

− 4π2α

∫ t

0

∑

ℓ

W (ℓ) ρ2(τ, ℓ) h1(τ, k − ℓ, η + k(t− τ)

)ℓ ·(η + k(t− τ)

)dτ

− 4π2α

∫ t

0

∑

ℓ

W (ℓ) ρ1(τ, ℓ) h2(τ, k − ℓ, η + k(t− τ)

)ℓ ·(η + k(t− τ)

)dτ,

ρ2(t, k) = −4π2α W (k)

∫ t

0

ρ2(τ, k) f 0(k(t− τ)

)|k|2 (t− τ)

)dτ(14.11)

− 4π2α

∫ t

0

∑

ℓ

W (ℓ) ρ1(τ, ℓ) h1(τ, k − ℓ, k(t− τ)

)ℓ · k (t− τ) dτ

− 4π2α

∫ t

0

∑

ℓ

W (ℓ) ρ2(τ, ℓ) h1(τ, k − ℓ, k(t− τ)

)ℓ · k (t− τ) dτ

− 4π2α

∫ t

0

∑

ℓ

W (ℓ) ρ1(τ, ℓ) h2(τ, k − ℓ, k(t− τ)

)ℓ · k (t− τ) dτ.

Here k and ℓ run over Zd.From (14.8)–(14.9) we see that ρ1 and h1 depend linearly on ε, and

(14.12) ρ1(t, k) = ε[ϕk(kt) +O(α)

], h1(t, k, η) = ε

[ϕk(η + kt) +O(α)

].

Then from (14.10)–(14.11), ρ2 and h2 are O(ε2 α); so by plugging (14.12) in theseequations we obtain(14.13)

ρ2(t, k) = −4π2ε2 α

∫ t

0

∑

ℓ

W (ℓ)ϕℓ(ℓτ)ϕk−ℓ(kt−ℓτ) ℓ·k (t−τ) dτ +O(ε2 α2)+O(ε3 α),

(14.14)

h2(t, k, η) = −4π2ε2 α

∫ t

0

∑

ℓ

W (ℓ)ϕℓ(ℓτ)ϕk−ℓ(η+kt−ℓτ) ℓ·(η+k(t−τ)

)dτ +O(ε2 α2)+O(ε3 α).


We plug these bounds again in the right-hand side of (14.10) to find

(14.15) h2(t, 0, η) = (II)ε(t, η) + (III)ε(t, η) +O(ε3 α3),

where

(II)ε(t, η) = −4π2 α

∫ t

0

∑

ℓ

(ℓ · η) W (ℓ) ρ1(τ, ℓ) h1(τ,−ℓ, η) dτ

is quadratic in ε, and (III)ε(t, η) is a third-order correction:

(III)ε(t, η) =16π4 ε3 α2∑

m,ℓ∈Zd

W (ℓ) W (m)

(14.16)

∫ t

0

∫ τ

0

ϕm(ms)ϕℓ−m(ℓτ −ms)ϕ−ℓ(η − ℓτ) (ℓ ·m) (τ − s)

+ ϕℓ(ℓτ)ϕ−ℓ−m(η − ℓτ −ms)m · (η − ℓ(τ − s))

(ℓ · η) ds dτ.

If f 0 is even, changing ϕk into ϕk(− · ) and η into −η amounts to change k into−k at the level of (14.8)–(14.9); but then (II)ε is invariant under this operation.We conclude that f is always homoclinic at second order in ε, and we consider theinfluence of the third-order term (14.16). Let

C[ϕ](η) := limt→∞

(III)ε(t, η).

After some relabelling, we find

(14.17) C[ϕ](η) = 16π4 ε3 α2∑

k,ℓ∈Zd

W (k) W (ℓ)

∫ ∞

0

∫ t

0

ϕℓ(ℓτ)ϕk−ℓ(kt− ℓτ)ϕ−k(η − kt) (k · ℓ) (t− τ)

+ ϕk(kt)ϕ−k−ℓ(η − kt− ℓτ) ℓ · (η − k(t− τ))

(k · η) dτ dt.

Now assume that ϕ−k = σ ϕk with σ = ±1. (σ = 1 means that the perturbation iseven in x; σ = −1 that it is odd.) Using the symmetry (k, ℓ) ↔ (−k,−ℓ) one cancheck that

C[ϕ S] S = σ C[ϕ],

where S(z) = −z. In particular, if the perturbation is odd in x, then the third-ordercorrection imposes a heteroclinic behavior for the solution, as soon as C[ϕ] 6= 0.


To construct an example where C[ϕ] 6= 0, we set d = 1, f 0 = Gaussian, fi − f 0 =

sin(2πx) θ1(v)+sin(4πx) θ2(v), ϕ1 = −ϕ−1 = θ1/2, ϕ2 = −ϕ−2 = θ2/2. The six pairs(k, ℓ) contributing to (14.17) are (−1, 1), (1,−1), (1, 2), (2, 1), (−1,−2), (−2,−1),

By playing on the respective sizes of W (1) and W (2) (which amounts in fact tochanging the size of the box), it is sufficient to consider the terms with coefficient

W (1)2, i.e., the pairs (−1, 1) and (1,−1). Then the corresponding bit of C[ϕ](η) is

−16π4 ε3 α2 W (1)2 η

∫ ∞

0

∫ t

0

[ϕ1(τ)ϕ1(η + t)ϕ2(−t+ τ) (t− τ)

+ ϕ1(τ)ϕ1(t)ϕ2(η + t− τ) (η + t− τ)

+ ϕ1(−τ)ϕ1(η − t)ϕ2(t+ τ) (t− τ)

+ ϕ1(−τ)ϕ1(t)ϕ2(η − t+ τ) (t− τ − η)]dτ dt.

If we let ϕ1 and ϕ2 vary in such a way that they become positive and almost con-centrated on R+, the only remaining term is the one in ϕ1(τ)ϕ2(η+ t−τ)ϕ1(t), andits contribution is negative for η > 0. So, at least for certain values of W (1) andW (2) there is a choice of analytic functions ϕ1 and ϕ2, such that C[ϕ] 6= 0. Thisdemonstrates the existence of heteroclinic trajectories.

To summarize: At first order in ε, the convergence is to the spatial average; atsecond order there is a homoclinic correction; at third order, if at least three modeswith zero sum are excited, there is possibility of heteroclinic behavior.

Remark 14.1. As pointed out to us by Bouchet, the existence of heteroclinic trajec-tories implies that the asymptotic behavior cannot be predicted on the basis of theinvariants of the equation and the interaction; indeed, the latter do not distinguishbetween the forward and backward solutions.

15. Beyond Landau damping

We conclude this paper with some general comments about the physical implica-tions of Landau damping.

Remark 14.1 show in particular that there is no “universal” large-time behaviorof the solution of the nonlinear Vlasov equation in terms of just, say, conservationlaws and the initial datum; the dynamics also have to enter explicitly. One canalso interpret this as a lack of ergodicity: the nonlinearity is not sufficient to makethe system explore the space of all “possible” distributions and to choose the mostfavorable one, whatever this means. Failure of ergodicity for a system of finitelymany particles was already known to occur, in relation to the KAM theorem; this


is mentioned e.g. in [60, p. 257] for the vortex system. There it is hoped that suchbehavior disappears as the dimension goes to infinity; but now we see that it alsoexists even in the infinite-dimensional setting of the Vlasov equation.

At first, this seems to be bad news for the statistical theory of the Vlasov equation,pioneered by Lynden-Bell [56] and explored by various authors [16, 63, 78, 88, 94],since even the sophisticated variants of this theory try to predict the likely finalstates in terms of just the characteristics of the initial data. In this sense, ourresults provide support for an objection raised by Isichenko [41, p. 2372] against thestatistical theory.

However, looking more closely at our proofs and results, proponents of the statis-tical theory will have a lot to rejoice about.

To start with, our results are the first to rigorously establish that the nonlinearVlasov equation does enjoy some asymptotic “stabilization” property in large time,without the help of any extra diffusion or ensemble averaging.

Next, the whole analysis is perturbative: each stable spatially homogeneous distri-bution will have its small “basin of damping”, and it may be that some distributionsare “much more stable” than others, say in the sense of having a larger basin.

Even more importantly, in Section 7 we have crucially used the smoothness toovercome the potentially destabilizing nonlinear effects. So any theory based on non-smooth functions might not be constrained by Landau damping. This certainly ap-plies to a statistical theory, for which smooth functions should be a zero-probabilityset.

Finally, to overcome the nonlinearity, we had to cope with huge constants (evenqualitatively larger than those appearing in classical KAM theory). If one believesin the explanatory virtues of proofs, these large constants might be the indicationthat Landau damping is a thin effect, which might be neglected when it comes topredict the “final” state in a “turbulent” situation.

Further work needs to be done to understand whether these considerations applyequally to the electrostatic and gravitational cases, or whether the electrostatic caseis favored in these respects; and what happens in “low” regularity.

Although the underlying mathematical and physical mechanisms differ, nonlinearLandau damping (as defined by Theorem 2.6) may arguably be to the theory ofVlasov equation what the KAM theorem is to the theory of Hamiltonian systems.


Like the KAM theorem, it might be conceptually important in theory and practice,and still be severely limited.18

Beyond the range of application of KAM theory lies the softer, more robust weakKAM theory developed by Fathi [26] in relation to Aubry–Mather theory. By a nicecoincidence, a Vlasov version of the weak KAM theory has just been developed byGangbo and Tudorascu [29], although with no relation to Landau damping. Makingthe connection is just one of the many developments which may be explored in thefuture.

Appendix

In this appendix we gather some elementary tools, our conventions, and somereminders about calculus. We write N0 = 0, 1, 2, . . ..

A.1. Calculus in dimension d. If n ∈ Nd0 we define

n! = n1! . . . nd!

and( n

m

)=( n1

m1

). . .( nd

md

).

If z ∈ Cd and n ∈ Zd, we let

‖z‖ = |z1| + . . .+ |zd|; zn = zn11 . . . znd

n ∈ C; |z|n = |zn|.In particular, if z ∈ Cd we have

e‖z‖ = e|z1|+...+|zd| =∑

n∈Nd0

‖z‖n

n!.

We may write e|z| instead of e‖z‖.

A.2. Multi-dimensional differential calculus. The Leibniz formula for func-tions f, g : R → R is

(fg)(n) =∑

m≤n

( nm

)f (m)g(n−m),

18It is a well-known scientific paradox that the KAM theorem was at the same time tremendouslyinfluential in the science of the twentieth century, and so restrictive that its assumptions areessentially never satisfied in practice.


where of course f (n) = dnf/dxn. The expression of derivatives of composed functionsis given by the Faa di Bruno formula:

(f G)(n) =∑

Pjmj=n

n!

m1! . . .mn!

(f (m1+...+mn) G

) n∏

j=1

(G(j)

j!

)mj

.

These formulas remain valid in several dimensions, provided that one defines, fora multi-index n = (n1, . . . , nd),

f (n) =∂n1

∂xn11

. . .∂nd

∂xndd

f.

They also remain true if (∂1, . . . , ∂d) is replaced by a d-tuple of commuting deriva-tion operators.

As a consequence, we shall establish the following Leibniz-type formula for oper-ators that are combinations of gradients and multiplications.

Lemma A.1. Let f and g be functions of v ∈ Rd, and a, b ∈ Cd. Then for anyn ∈ Nd,

(∇v + (a+ b))n(fg) =∑

m≤n

( nm

)(∇v + a)mf (∇v + b)n−mg.

Proof. The right-hand side is equal to∑

m,q,r

( nm

)( mq

)( n − mr

)∇q

vf ∇rvg a

m−q bn−m−r.

After changing indices p = q + r, s = m− q, this becomes∑

s,p,r

( np

)( pr

)( n − ps

)∇r

vg∇p−rv f as bn−p−s =

∑

p

( np

)∇p

v(fg) (a+ b)n−p

= (∇v + (a+ b))n(fg).

A.3. Fourier transform. If f is a function Rd → R, we define

(A.1) f(η) =

∫

Rd

e−2iπη·v f(v) dv;

then we have the usual formulas

f(v) =

∫

Rd

f(η) e2iπη·v dη; ∇f(η) = 2iπη f(η).


Let TdL = Rd/(LZd). If f is a function Td

L → R, we define

(A.2) f (L)(k) =

∫

TdL

e−2iπ kL·x f(x) dx;

then we have

f(x) =1

Ld

∑

k∈Zd

f (L)(k) e2iπ kL

x; ∇f(L)

(k) = 2iπk

Lf (L)(k).

If f is a function TdL × Rd → R, we define

(A.3) f (L)(k, η) =

∫

TdL

∫

Rd

e−2iπ kL·x e−2iπη·v f(x, v) dx dv;

so that the reconstruction formula reads

f(x, v) =1

Ld

∑

k∈Zd

∫

Rd

f (L)(k, η) e2iπ kL·xe2iπη·v dv.

When L = 1 we do not specify it: so we just write

f = f (1); f = f (1).

(There is no risk of confusion since in that case, (A.3) and (A.1) coincide.)

A.4. Fixed point theorem. The following theorem is one of the many variantsof the Picard fixed point theorem. We write B(0, R) for the ball of center 0 andradius R.

Theorem A.2 (Fixed point theorem). Let E be a Banach space, F : E → E, andR = 2‖F (0)‖. If F is (1/2)-Lipschitz B(0, R) → E, then it has a unique fixed pointin B(0, R).

Proof. Uniqueness is obvious. To prove existence, run the classical Picard iterativescheme initialized at 0: x0 = 0, x1 = F (0), x2 = F (F (0)), etc. It is clear that (xn) isa Cauchy sequence and ‖xn‖ ≤ ‖F (0)‖(1 + . . .+ 1/2n) ≤ 2‖F (0)‖, so xn convergesin B(0, R) to a fixed point of F .

A.5. Plemelj formula. The Plemelj formula states that, in D′(R),

(A.4)1

x− i0= p.v.

(1

x

)+ iπ δ0;


or equivalently,

(A.5)1

x+ i0= p.v.

(1

x

)− iπ δ0.

Let us give a complete proof of this formula which plays a crucial role in plasmaphysics.

Proof of (A.4). First we recall that

p.v.

(1

x

)= lim

ε→0

1|x|≥ε

x.

A more explicit expression can be given for this limit. Let χ be an even functionon R with χ(0) = 1, χ ∈ Lip ∩ L1(R); in particular

∫1|x|≥ε χ(x) dx/x = 0 and

(1 − χ(x))/x is bounded. Then for any ϕ ∈ Lip(R) ∩ L1(R),∫

|x|≥ε

ϕ(x)

xdx =

∫

|x|≥ε

ϕ(x)

xχ(x) dx+

∫

|x|≥ε

ϕ(x)

x(1 − χ(x)) dx(A.6)

=

∫

|x|≥ε

[ϕ(x) − ϕ(0)

x

]χ(x) dx+

∫

|x|≥ε

ϕ(x)(1 − χ(x))

xdx

−−→ε→0

∫ (ϕ(x) − ϕ(0)

x

)χ(x) dx+

∫ϕ(x)

(1 − χ(x))

xdx.

Then (A.4) can be rewritten in the following more explicit way: for any ϕ ∈ Lip(R)∩L1(R) and any χ satisfying the above assumptions,

(A.7) limλ→0+

∫

R

ϕ(x)

x− iλdx =

∫ (ϕ(x) − ϕ(0)

x

)χ(x) dx+

∫ϕ(x)

(1 − χ(x))

xdx

+ iπϕ(0).

Now, to prove (A.7), let us pick such a function χ and write, for λ > 0,

∫ϕ(x)

x− iλdx =

∫ (ϕ(x) − ϕ(0)

x− iλ

)χ(x) dx+

∫ϕ(x)

(1 − χ(x))

x− iλdx

+ ϕ(0)

(∫χ(x)

x− iλdx

).

As λ → 0, the first two integrals on the right-hand side converge to the right-handside of (A.6), and there is an extra term proportional to ϕ(0); so it only remains to


check that

(A.8)

∫χ(x)

x− iλdx −−−→

λ→0+iπ.

If (A.8) holds for some particular χ satisfying the requested conditions, then (A.4)follows and it implies that (A.8) holds for any such χ. So let us pick one particular

χ, say χ(x) = e−x2, which can be extended throughout the complex plane into a

holomorphic function. Then since the complex integral is invariant under contourdeformation, ∫

R

e−x2

x− iλdx =

∫

Cε

e−z2

z − iλdz,

where Cε is the complex contour made of the straight line (−∞,−ε), followed by thehalf-circleDε = −εeiθ0≤θ≤π, followed by the straight line (ε,∞). The contributionsof both straight lines cancel each other by symmetry, and only remains the integralon the half-circle Dε. As λ → 0 this integral approaches

∫Dεe−z2

dz/z, which as

ε→ 0 becomes close to∫

Dεdz/z = iπ, as was claimed.

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Clement Mouhot

University of CambridgeDPMMS, Centre for Mathematical Sciences

Wilberforce RoadCambridge CB3 0WA

UK

e-mail: [email protected]

Cedric Villani

ENS Lyon & Institut Universitaire de FranceUMPA, UMR CNRS 5669

46 allee d’Italie69364 Lyon Cedex 07

FRANCE

e-mail: [email protected]

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ON LANDAU DAMPING - Cédric Villani LANDAU DAMPING C. MOUHOT AND C. VILLANI Abstract. Going beyond the linearized study has been a longstanding problem in the theory of Landau damping.Published